# questions

title: for-an-insulator-the-forbidden-energy-gap-is
slug: for-an-insulator-the-forbidden-energy-gap-is-14187
question: <div class="question">For an insulator, the forbidden energy gap is</div>
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>Zero
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$1 \mathrm{eV}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$2 \mathrm{eV}$
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>$5 \mathrm{eV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$5 \mathrm{eV}$</span> </div>

answer:
<div class="solution">For an insulator, $E_{\mathrm{g}}&gt;3 \mathrm{eV}$, that is why electron transition from valence band to conduction band is not possible. For semiconductor $E_{\mathrm{g}}$ is $0.2 \mathrm{eV}$ to $0.3 \mathrm{eV}$, while for metals $E_g&gt;5 \mathrm{eV}$.</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P2.db
title:
for-an-isomerisation-reaction-a-b-the-temperature-dependence-of-equilibrium-constant-is-given-by-lo-g-e-k-40-t-2000-the-value-of-s-at-hook-is-therefor

slug:
for-an-isomerisation-reaction-a-b-the-temperature-dependence-of-equilibrium-constant-is-given-by-lo-g-e-k-40-t-2000-the-value-of-s-at-hook-is-therefor-50205

question:
<div class="question">For an isomerisation reaction $A \rightleftharpoons B$, the temperature dependence of equilibrium constant is given by
<br/>$\log _e K=4.0-\frac{2000}{T}$
<br/>The value of $\Delta S^{\circ}$ at Hook is, therefore</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$4 R$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$5 R$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$400 R$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$2000 R$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$4 R$
<br/></span> </div>

answer:
<div class="solution">van't Hoff equation giving variation of equilibrium constant, $K$ with temperature $T$ is
<br/>$\log _e K=-\frac{\Delta H^{\circ}}{R T}+\frac{\Delta S^{\circ}}{R}$
<br/>Given, $\quad \log _e K=-\frac{2000}{T}+4.0$
<br/>$\therefore \quad \Delta S^{\circ} / R=4 \Rightarrow \Delta S^{\circ}=4 R$</div>


title: for-an-isothermal-expansion-of-a-perfect-gas-the-value-of-p-p-is-equal-to
slug: for-an-isothermal-expansion-of-a-perfect-gas-the-value-of-p-p-is-equal-to-17963
tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch1_P2.db
question:
<div class="question">For an isothermal expansion of a perfect gas, the value of $\frac{\Delta P}{P}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\gamma^{1 / 2} \frac{\Delta V}{V}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{\Delta V}{V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\gamma \frac{\Delta V}{V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\gamma^{2} \frac{\Delta V}{V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{\Delta V}{V}$</span> </div>

answer:
<div class="solution">Differentiate $P V=$ constant w.r.t $V$
<br/>$\Rightarrow P \Delta V+V \Delta P=0 \quad \Rightarrow \frac{\Delta P}{P}=-\frac{\Delta V}{V}$</div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch1_P2.db
title:
for-an-rlc-circuit-driven-with-voltage-of-amplitude-v-m-and-frequency-0-lc-1-the-current-exhibits-resonance-the-quality-factor-q-is-given-by

slug:
for-an-rlc-circuit-driven-with-voltage-of-amplitude-v-m-and-frequency-0-lc-1-the-current-exhibits-resonance-the-quality-factor-q-is-given-by-77174

question:
<div class="question">For an RLC circuit driven with voltage of amplitude $V_{m}$ and frequency $\omega_{0}=\frac{1}{\sqrt{\mathrm{LC}}}$ the current exhibits resonance the quality factor, $Q$ is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\omega_{0}}{\mathrm{~L}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\omega_{0} \mathrm{~L}}{\mathrm{R}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{R}}{\omega_{0} \mathrm{C}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{CR}}{\omega_{0}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\omega_{0} \mathrm{~L}}{\mathrm{R}}$</span> </div>

answer:
<div class="solution">Total angular momentum of the system initially \(\mathrm{L}_{\mathrm{i}}=\mathrm{I}_{1} \mathrm{w}+\mathrm{I}_{2}(0)=\mathrm{I}_{1} \mathrm{w}\) Total angular momentum of the system finally \(\mathrm{L}_{\mathrm{f}}=\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \mathrm{w}_{2}\)
<br/> <br/>According to conservation of angular momentum i.e. \(\mathrm{L}_{\mathrm{i}}=\mathrm{L}_{\mathrm{f}}\)
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\Rightarrow \mathrm{I}_{1} \omega=\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \omega_{2} \\
<br/> <br/>&amp;\Rightarrow \omega_{2}=\frac{\mathrm{I}_{1} \omega}{\mathrm{I}_{1}+\mathrm{I}_{2}}
<br/> <br/>\end{aligned}\)</div>


title: for-any-differentiable-function-y-of-x-d-y-2-d-2-x-d-x-d-y-3-d-x-2-d-2-y
slug: for-any-differentiable-function-y-of-x-d-y-2-d-2-x-d-x-d-y-3-d-x-2-d-2-y-67679
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch1_P2.db
question:
<div class="question">For any differentiable function $y$ of $x$, $\frac{d^{2} x}{d y^{2}}\left(\frac{d y}{d x}\right)^{3}+\frac{d^{2} y}{d x^{2}}=$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-y$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x$</span> </li> </ul>

answer:
<div class="solution">$\frac{d y}{d x}=\left(\frac{d x}{d y}\right)^{-1}$
<br/> <br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}=-1\left(\frac{d x}{d y}\right)^{-2}\left\{\frac{d}{d x}\left(\frac{d x}{d y}\right)\right\}$
<br/> <br/>$\Rightarrow \frac{d^{2} y}{d x^{2}}=(-1)\left(\frac{d x}{d y}\right)^{-2}\left\{\frac{d}{d y}\left(\frac{d x}{d y}\right) \frac{d y}{d x}\right\}$
<br/> <br/>$=(-1)\left(\frac{d y}{d x}\right)^{2}\left\{\frac{d^{2} x}{d y^{2}} \cdot \frac{d y}{d x}\right\}=-\left(\frac{d y}{d x}\right)^{3}\left\{\frac{d^{2} x}{d y^{2}}\right\}$
<br/> <br/>$\Rightarrow \frac{d^{2} x}{d y^{2}}\left(\frac{d y}{d x}\right)^{3}+\frac{d^{2} y}{d x^{2}}=0$</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch1_P2.db
title:
for-each-parabola-y-x-2-p-x-q-meeting-coordinate-axes-at-3distinct-points-if-circles-are-drawn-through-these-points-then-the-family-of-circles-must-pa

slug:
for-each-parabola-y-x-2-p-x-q-meeting-coordinate-axes-at-3distinct-points-if-circles-are-drawn-through-these-points-then-the-family-of-circles-must-pa-24552

question:
<div class="question">For each parabola $y=x^2+p x+q$, meeting coordinate axes at 3-distinct points, if circles are drawn through these points, then the family of circles must pass through</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$(1,0)$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$(0,1)$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$(1,1)$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$(p, q)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$(0,1)$
<br/></span> </div>

answer:
<div class="solution">Suppose the parabola $y=x^2+p x+q$ cuts $X$-axis at $A(\alpha, 0)$ and $B(\beta, 0)$.
<br/>Then, $\alpha, \beta$ are roots of the equation $x^2+p x+q=0$
<br/>$$
<br/>\therefore \quad \alpha+\beta=-p \text { and } \alpha \beta=q
<br/>$$
<br/>The parabola $y=x^2+p x+q$ cuts $Y$-axis at $(0, q)$.
<br/>Let the equation of the circle passing through $A, B$ and $C$ be
<br/>$$
<br/>\begin{aligned}
<br/>&amp; x^2+y^2+2 g x+2 f y+c  =0 &amp;...(i)\\
<br/>\therefore &amp; \alpha^2+2 g \alpha+c  =0 &amp;...(ii)\\
<br/>&amp; \beta^2+2 g \beta+c  =0 &amp;...(iii)\\
<br/>\text { and } &amp; q^2+2 f q+c  =0 &amp;...(iv)
<br/>\end{aligned}
<br/>$$
<br/>Subtracting Eq. (iii) from Eq. (ii), we get
<br/>$$
<br/>\begin{array}{rlrl}
<br/>\alpha+\beta+2 g &amp; =0 \\
<br/>\Rightarrow &amp; g = p / 2
<br/>\end{array}
<br/>$$
<br/>Adding Eqs. (ii) and (iii), we get
<br/>$$
<br/>\begin{array}{r}
<br/>\alpha^2+\beta^2+2 g(\alpha+\beta)+2 c=0 \\
<br/>(\alpha+\beta)^2-2 \alpha \beta+2 g(\alpha+\beta)+2 c=0 \\
<br/>p^2-2 q-p^2+2 c=0
<br/>\end{array}
<br/>$$
<br/>$$
<br/>\left[\because \alpha+\beta=p \text { and } g=\frac{p}{2}\right]
<br/>$$
<br/>Putting $c=q$ in Eq. (iv), we get $f=-\left(\frac{q+1}{2}\right)$
<br/>Substituting the values of $g, f$ and $c$ in Eq. (i), we obtain the equation of family of circles passing through $A, B$ and $C$ as
<br/>$$
<br/>x^2+y^2+p x-(q+1) y+q=0
<br/>$$
<br/>Clearly, it passes through $(0,1)$.</div>


title: if-1-tan-tan-1-1-tan-tan-1-1-a-b-b-a-then
slug: if-1-tan-tan-1-1-tan-tan-1-1-a-b-b-a-then-79207
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">$\text { If }\left[\begin{array}{cc}
<br/>1 &amp; -\tan \theta \\
<br/>\tan \theta &amp; 1
<br/>\end{array}\right]\left[\begin{array}{cc}
<br/>1 &amp; \tan \theta \\
<br/>-\tan \theta &amp; 1
<br/>\end{array}\right]^{-1}=\left[\begin{array}{cc}
<br/>a &amp; -b \\
<br/>b &amp; a
<br/>\end{array}\right],$ 
<br/>then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$a=1, b=1$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$a=\sin 2 \theta, b=\cos 2 \theta$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$a=\cos 2 \theta, b=\sin 2 \theta$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$a=\cos 2 \theta, b=\sin 2 \theta$
<br/></span> </div>

answer:
<div class="solution">Given,
<br/>$\left[\begin{array}{cc}1 &amp; -\tan \theta \\ \tan \theta &amp; 1\end{array}\right]\left[\begin{array}{cc}1 &amp; \tan \theta \\ -\tan \theta &amp; 1\end{array}\right]^{-1}=\left[\begin{array}{cc}a &amp; -b \\ b &amp; a\end{array}\right]$
<br/>$\Rightarrow\left[\begin{array}{cc}1 &amp; -\tan \theta \\ \tan \theta &amp; 1\end{array}\right] \cdot \frac{1}{1+\tan ^2 \theta}\left[\begin{array}{cc}1 &amp; -\tan \theta \\ \tan \theta &amp; 1\end{array}\right]$
<br/>$=\left[\begin{array}{cc}a &amp; -b \\ b &amp; a\end{array}\right]$
<br/>$\Rightarrow \frac{1}{1+\tan ^2 \theta}\left[\begin{array}{cc}1-\tan ^2 \theta &amp; -2 \tan \theta \\ 2 \tan \theta &amp; 1-\tan ^2 \theta\end{array}\right]=\left[\begin{array}{cc}a &amp; -b \\ b &amp; a\end{array}\right]$
<br/>$\Rightarrow \quad\left[\begin{array}{cc}\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta} &amp; \frac{2 \tan \theta}{1+\tan ^2 \theta} \\ \frac{2 \tan \theta}{1+\tan ^2 \theta} &amp; \frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}\end{array}\right]=\left[\begin{array}{cc}a &amp; -b \\ b &amp; a\end{array}\right]$
<br/>$\Rightarrow\left[\begin{array}{cc}\cos 2 \theta &amp; -\sin 2 \theta \\ \sin 2 \theta &amp; \cos 2 \theta\end{array}\right]=\left[\begin{array}{cc}a &amp; -b \\ b &amp; a\end{array}\right]$
<br/>$\Rightarrow \quad a=\cos 2 \theta, b=\sin 2 \theta$</div>


title: if-1-x-x-2-20-r-0-40-a-r-x-r-then-r-0-39-1-r-a-r-a-r-1-equal-to
slug: if-1-x-x-2-20-r-0-40-a-r-x-r-then-r-0-39-1-r-a-r-a-r-1-equal-to-78275
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>20</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>40</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><mo>·</mo><msup><mi>x</mi><mi>r</mi></msup></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>39</mn></munderover><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mi>r</mi></msup><mo>·</mo><msub><mi>a</mi><mi>r</mi></msub><mo>·</mo><msub><mi>a</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math> equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>79</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>39</mn></msup><mo>·</mo><mmultiscripts><mi>C</mi><mn>39</mn><none></none><mprescripts></mprescripts><none></none><mn>78</mn></mmultiscripts></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mn>39</mn></msup><mo>·</mo><mmultiscripts><mi>C</mi><mn>39</mn><none></none><mprescripts></mprescripts><none></none><mn>78</mn></mmultiscripts></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>20</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>40</mn></munderover><msub><mi>a</mi><mi>r</mi></msub><msup><mi>x</mi><mi>r</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow></mfenced><mn>20</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>40</mn></munderover><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mi>r</mi></msup><msub><mi>a</mi><mi>r</mi></msub><msup><mfenced><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced><mi>r</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>20</mn></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>40</mn></munderover><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mi>r</mi></msup><msub><mi>a</mi><mi>r</mi></msub><msup><mi>x</mi><mrow><mn>40</mn><mo>-</mo><mi>r</mi></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mn>39</mn></munderover><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mi>r</mi></msup><msub><mi>a</mi><mi>r</mi></msub><mo>·</mo><msub><mi>a</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math> is coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in product of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>20</mn></msup><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow></mfenced><mn>20</mn></msup></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>20</mn></msup><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow></mfenced><mn>20</mn></msup><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow></mfenced><mn>20</mn></msup></math></p><p>Here, there will be no term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p><p>Hence, the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>.</p></div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
title:
if-02-gram-of-an-organic-compound-containing-carbon-hydrogen-and-oxygen-on-combustion-yielded-0147-gram-carbon-dioxide-and-012-gram-water-what-will-be

slug:
if-02-gram-of-an-organic-compound-containing-carbon-hydrogen-and-oxygen-on-combustion-yielded-0147-gram-carbon-dioxide-and-012-gram-water-what-will-be-34826

question:
<div class="question">If 0.2 gram of an organic compound containing carbon, hydrogen and oxygen on combustion, yielded 0.147 gram carbon dioxide and 0.12 gram water. What will be the content of oxygen in the substance?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$73.29 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$78.45 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$83.23 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$89.50 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$73.29 \%$</span> </div>

answer:
<div class="solution">$\%$ of $\mathrm{C}=\frac{12}{44} \times \frac{0.147}{0.2} \times 100=20.045 \%$
<br/> <br/>$\%$ of $\mathrm{H}=\frac{2}{18} \times \frac{0.12}{0.2} \times 100=6.666 \%$
<br/> <br/>$\%$ of $\mathrm{O}=100-(20.045+6.666)=73.29 \%$</div>


title: if-2-x-2-y-2-x-y-then-d-x-d-y
slug: if-2-x-2-y-2-x-y-then-d-x-d-y-81933
question: <div class="question">$\text { If } 2^{x}+2^{y}=2^{x+y} \text {, then } \frac{d y}{d x}=$</div>
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{x-y} \frac{2^{y}-1}{2^{x}-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2^{x-y} \frac{2^{y}-1}{1-2^{x}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2^{x}+2^{y}}{2^{x}-2^{y}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">On differentiating
<br/> <br/>$\begin{array}{l}
<br/> <br/>2^{x} \log 2+2^{y} \log 2 \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\
<br/> <br/>=2^{x} \cdot 2^{y} \frac{\mathrm{dy}}{\mathrm{dx}} \cdot \log 2+2^{y} \cdot 2^{x} \log 2 \\
<br/> <br/>\Rightarrow 2^{x}+2^{y} \frac{\mathrm{dy}}{\mathrm{dx}}=2^{x+y} \frac{\mathrm{dy}}{\mathrm{dx}}+2^{x+y} \\
<br/> <br/>\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2^{x+y}-2^{x}}{2^{y}-2^{x+y}} \\
<br/> <br/>\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2^{x}+2^{y}-2^{x}}{2^{y}-2^{x}-2^{y}}=-2^{y-x}
<br/> <br/>\end{array}$</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-4-5-is-one-vertex-and-7-x-y-8-0-is-one-diagonal-of-a-square-then-the-equation-of-second-diagonal-is-1

slug:
if-4-5-is-one-vertex-and-7-x-y-8-0-is-one-diagonal-of-a-square-then-the-equation-of-second-diagonal-is-1-12629

question:
<div class="question">If $(-4,5)$ is one vertex and $7 x-y+8=0$ is one diagonal of a square, then the equation of second diagonal is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+3 y=21$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x-3 y=7$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x+7 y=31$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x+3 y=21$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+7 y=31$</span> </div>

answer:
<div class="solution">One vertex of square is $(-4,5)$ and equation of one diagonal is $7 x-y+8=0$ Diagonal of a square are perpendicular and bisect each other<br/>Let the equation of the other diagonal be $y=m x+$ $c$ where $m$ is the slope of the line and $c$ is the $\mathrm{y}$-intercept.<br/>Since this line passes through $(-4,5)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rx_M4u4E1F-XYhQ1xJqFBFhP7bKLFiyVP_HtPQa9Gk0.original.fullsize.png"/><br/><br/>Since this line is at right angle to the line<br/>$7 x-y+8=0$ or $y=7 x+8$, having slope $=7$,<br/>$\therefore 7 \times \mathrm{m}=-1$ or $\mathrm{m}=\frac{-1}{7}$<br/>Putting this value of $m$ in equation (i) we get<br/>$5=-4 \times\left(\frac{-1}{7}\right)+c$<br/>or $5=\frac{4}{7}+c \quad$ or $\quad c=5-\frac{4}{7}=\frac{31}{7}$<br/>Hence equation of the other diagonal is<br/>$y=-\frac{1}{7} x+\frac{31}{7}$ or $7 y=-x+31$ or $x+7 y-31=0$ or $x+7 y=31$</div>


tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-4-5-is-one-vertex-and-7-x-y-8-0-is-one-diagonal-of-a-square-then-the-equation-of-second-diagonal-is-3

slug:
if-4-5-is-one-vertex-and-7-x-y-8-0-is-one-diagonal-of-a-square-then-the-equation-of-second-diagonal-is-3-67738

question:
<div class="question">If $(-4,5)$ is one vertex and $7 x-y+8=0$ is one diagonal of a square, then the equation of second diagonal is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+3 y=21$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 x-3 y=7$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x+7 y=31$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x+3 y=21$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+7 y=31$</span> </div>

answer:
<div class="solution">One vertex of square is $(-4,5)$ and equation of one diagonal is $7 x-y+8=0$ Diagonal of a square are perpendicular and bisect each other<br/>Let the equation of the other diagonal be $y=m x+$ $c$ where $m$ is the slope of the line and $c$ is the $\mathrm{y}$-intercept.<br/>Since this line passes through $(-4,5)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rx_M4u4E1F-XYhQ1xJqFBFhP7bKLFiyVP_HtPQa9Gk0.original.fullsize.png"/><br/><br/>Since this line is at right angle to the line<br/>$7 x-y+8=0$ or $y=7 x+8$, having slope $=7$,<br/>$\therefore 7 \times \mathrm{m}=-1$ or $\mathrm{m}=\frac{-1}{7}$<br/>Putting this value of $m$ in equation (i) we get<br/>$5=-4 \times\left(\frac{-1}{7}\right)+c$<br/>or $5=\frac{4}{7}+c \quad$ or $\quad c=5-\frac{4}{7}=\frac{31}{7}$<br/>Hence equation of the other diagonal is<br/>$y=-\frac{1}{7} x+\frac{31}{7}$ or $7 y=-x+31$ or $x+7 y-31=0$ or $x+7 y=31$</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
if-4-a-2-b-2-2-c-2-4-ab-6-a-c-3-b-c-0-the-family-of-lines-ax-by-c-0-is-concurrent-at-one-or-the-other-of-the-two-points

slug:
if-4-a-2-b-2-2-c-2-4-ab-6-a-c-3-b-c-0-the-family-of-lines-ax-by-c-0-is-concurrent-at-one-or-the-other-of-the-two-points-84575

question:
<div class="question">$\quad$ If $4 a^{2}+b^{2}+2 c^{2}+4 a b-6 a c-3 b c=0,$ the family of lines $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ is concurrent at one or the other of the two points-</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(-1,-\frac{1}{2}\right),(-2,-1)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-1,-1),\left(-2,-\frac{1}{2}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-1,2),\left(\frac{1}{2},-1\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(1,2),\left(\frac{1}{2},-1\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(-1,-\frac{1}{2}\right),(-2,-1)$</span> </div>

answer:
<div class="solution">$$
<br/> <br/>\text {  } \begin{array}{l}
<br/> <br/>4 a^{2}+b^{2}+2 c^{2}+4 a b-6 a c-3 b c \\
<br/> <br/>\quad \equiv(2 a+b)^{2}-3(2 a+b) c+2 c^{2}=0 \\
<br/> <br/>\quad \Rightarrow(2 a+b-2 c)(2 a+b-c)=0 \Rightarrow c=2 a+b \\
<br/> <br/>\quad \text { or } c=a+\frac{1}{2} b
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>The equation of the family of lines is
<br/> <br/>$$
<br/> <br/>a(x+2)+b(y+1)=0 \text { or } a(x+1)+b\left(y+\frac{1}{2}\right)=0
<br/> <br/>$$
<br/> <br/>giving the point of concurrence (-2,-1) or
<br/> <br/>$$
<br/> <br/>\left(-1,-\frac{1}{2}\right)
<br/> <br/>$$</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
if-10-mole-of-i-2-is-introduced-into-10-litre-flask-at-1000-k-at-equilibrium-k-c-1-0-6-which-one-is-correct-1

slug:
if-10-mole-of-i-2-is-introduced-into-10-litre-flask-at-1000-k-at-equilibrium-k-c-1-0-6-which-one-is-correct-1-45974

question:
<div class="question">If 1.0 mole of $I_{2}$ is introduced into 1.0 litre flask at $1000 \mathrm{~K},$ at equilibrium $\left(\mathrm{K}_{\mathrm{c}}=10^{-6}\right),$ which one is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]&gt;\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]&lt;\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]=\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]=\frac{1}{2}\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]&lt;\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/JUMljcBOs8oao9k61hYcukGCiES4JNR-uo4gF1JVF6w.original.fullsize.png"/><br/> <br/> <br/>$K_{C}=\frac{(2 x)^{2}}{(1-x)}=10^{-6}$
<br/> <br/>It shows that $(1-x)&lt;2 x$</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
if-10-mole-of-i-2-is-introduced-into-10-litre-flask-at-1000-k-at-equilibrium-k-c-1-0-6-which-one-is-correct

slug:
if-10-mole-of-i-2-is-introduced-into-10-litre-flask-at-1000-k-at-equilibrium-k-c-1-0-6-which-one-is-correct-89884

question:
<div class="question">If 1.0 mole of $I_{2}$ is introduced into 1.0 litre flask at $1000 \mathrm{~K},$ at equilibrium $\left(\mathrm{K}_{\mathrm{c}}=10^{-6}\right),$ which one is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]&gt;\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]&lt;\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]=\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]=\frac{1}{2}\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{I}_{2}(\mathrm{~g})\right]&lt;\left[\mathrm{I}^{-}(\mathrm{g})\right]$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/JUMljcBOs8oao9k61hYcukGCiES4JNR-uo4gF1JVF6w.original.fullsize.png"/><br/> <br/> <br/>$K_{C}=\frac{(2 x)^{2}}{(1-x)}=10^{-6}$
<br/> <br/>It shows that $(1-x)&lt;2 x$</div>


title: if-12-cot-2-theta31-operatornamecoses-theta320-then-the-value-of-sin-theta-is
slug: if-12-cot-2-theta31-operatornamecoses-theta320-then-the-value-of-sin-theta-is-94909
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(12 \cot ^2 \theta-31 \operatorname{coses} \theta+32=0\), then the value of \(\sin \theta\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\frac{3}{5}\) or 1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{2}{3}\) or \(\frac{-2}{3}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{4}{5}\) or \(\frac{3}{4}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\pm \frac{1}{2}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{3}{5}\) or 1</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; 12 \cot ^2 \theta-31 \operatorname{cosec} \theta+32=0 \\<br/>&amp; \Rightarrow 12\left(\operatorname{cosec}^2 \theta-1\right)-31 \operatorname{cosec} \theta+32=0 \\<br/>&amp; \Rightarrow \quad 12 \operatorname{cosec}^2 \theta-31 \operatorname{cosec} \theta+20=0 \\<br/>&amp; \Rightarrow \quad 12 \operatorname{cosec}^2 \theta-16 \operatorname{cosec} \theta-15 \operatorname{cosec} \theta+20=0 \\<br/>&amp; \Rightarrow \quad(4 \operatorname{cosec} \theta-5)(3 \operatorname{cosec} \theta-4)=0 \\<br/>&amp; \Rightarrow \quad \operatorname{cosec} \theta=\frac{5}{4}, \frac{4}{3} ; \therefore \sin \theta=\frac{4}{5}, \frac{3}{4}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">35</span> </div>
source_db: MarksBatch2_P1.db
title:
if-16-identical-pencils-are-distributed-among-4-children-such-that-each-gets-at-least-3-pencils-the-number-of-ways-of-distributing-the-pencils-is

slug:
if-16-identical-pencils-are-distributed-among-4-children-such-that-each-gets-at-least-3-pencils-the-number-of-ways-of-distributing-the-pencils-is-95945

question:
<div class="question">If 16 identical pencils are distributed among 4 children such that each gets at least 3 pencils. The number of ways of distributing the pencils is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">15</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">25</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">35</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">40</span> </li> </ul>

answer:
<div class="solution">Required number of ways
<br/> <br/>$=\text { coeff. of } x^{16} \text { in }\left(x^{3}+x^{4}+x^{5}+x^{6}+x^{7}\right)^{4}$</div>


title: if-20-mathrmcmathrmr-20-mathrmcmathrmr10-then-18-mathrmcmathrmr-is-equal-to
slug: if-20-mathrmcmathrmr-20-mathrmcmathrmr10-then-18-mathrmcmathrmr-is-equal-to-14551
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">816</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If \({ }^{20} \mathrm{C}_{\mathrm{r}}={ }^{20} \mathrm{C}_{\mathrm{r}-10}\) then \({ }^{18} \mathrm{C}_{\mathrm{r}}\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4896</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">816</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1632</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">\({ }^{20} \mathrm{C}_{\mathrm{r}}={ }^{20} \mathrm{C}_{\mathrm{r}-10} \Rightarrow \mathrm{r}+(\mathrm{r}-10)=20 \Rightarrow \mathrm{r}=15\)<br/>\(\therefore \quad{ }^{18} \mathrm{C}_{\mathrm{r}}={ }^{18} \mathrm{C}_{15}={ }^{18} \mathrm{C}_3=\frac{18 \cdot 17 \cdot 16}{1 \cdot 2 \cdot 3}=816\)</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-1000-droplets-of-water-of-surface-tension-007-n-m-having-same-radius-1-mm-each-combine-to-from-a-single-drop-in-the-process-the-released-surface-en

slug:
if-1000-droplets-of-water-of-surface-tension-007-n-m-having-same-radius-1-mm-each-combine-to-from-a-single-drop-in-the-process-the-released-surface-en-59481

question:
<div class="question">If 1000 droplets of water of surface tension 0.07 $\mathrm{N} / \mathrm{m}$. having same radius $1 \mathrm{~mm}$ each, combine to from a single drop. In the process the released surface energy is( Take $\left.\pi=\frac{22}{7}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7.92 \times 10^{-6} \mathrm{~J}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$7.92 \times 10^{-4} \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$9.68 \times 10^{-4} \mathrm{~J}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$8.8 \times 10^{-5} \mathrm{~J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7.92 \times 10^{-4} \mathrm{~J}$</span> </div>

answer:
<div class="solution">We have $V_f=V_i$<br/>$\Rightarrow \frac{4}{3} \pi r_{\mathrm{f}}^3=1000 \times \frac{4}{3} \pi \mathrm{r}_{\mathrm{i}}^3 \Rightarrow \mathrm{r}_{\mathrm{f}}^3=1000 \mathrm{r}_{\mathrm{i}}^3$<br/>$\Rightarrow r_f=10 r_i$<br/>So, released energy $=$ Initial surface energy - final surface energy<br/>$=1000 \times \mathrm{T} \times 4 \pi \mathrm{r}_{\mathrm{i}}^2-\mathrm{T} \times 4 \pi \mathrm{r}_{\mathrm{f}}^2$<br/>$=4 \pi \mathrm{T}\left(1000 \mathrm{r}_{\mathrm{i}}^2-\mathrm{r}_{\mathrm{f}}^2\right)$<br/>$=4 \pi \times 0.07\left(1000 \mathrm{r}_{\mathrm{i}}^2-100 \mathrm{r}_{\mathrm{i}}^2\right)$<br/>$=4 \pi \times 0.07 \times 900 r_i^2$<br/>$=4 \pi \times 63 \times 10^{-6}=7.92 \times 10^{-4} \mathrm{~J}$</div>


title: if-a-0-3-2-4-and-k-a-0-2-b-3-a-24-then-the-values-of-k-a-and-b-respectively-are
slug: if-a-0-3-2-4-and-k-a-0-2-b-3-a-24-then-the-values-of-k-a-and-b-respectively-are-71947
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{cc}0 &amp; 2 \\ 3 &amp; -4\end{array}\right]$ and $k \mathrm{~A}=\left[\begin{array}{cc}0 &amp; 3 a \\ 2 b &amp; 24\end{array}\right]$, then the values of $k, a$ and $b$ respectively are:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> $-6,-12,-18$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-6,-4,-9$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-6,4,9$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-6,12,18$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-6,-4,-9$</span> </div>

answer:
<div class="solution">$k A=\left[\begin{array}{cc}0 &amp; 2 k \\ 3 k &amp; -4 k\end{array}\right]=\left[\begin{array}{cc}0 &amp; 3 a \\ 2 b &amp; 24\end{array}\right]$<br/>$\Rightarrow \quad k=-6, a=-4$ and $b=-9$</div>


title: if-a-0-a-in-r-za2-i-and-zza-z10-then
slug: if-a-0-a-in-r-za2-i-and-zza-z10-then-46992
question: <div class="question">If \(a &gt; 0, a \in R, z=a+2 i\) and \(z|z|-a z+1=0\) then</div>
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(z\) is always a positive real number</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(z\) is always a negative real number</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(z\) is purely imaginary number</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">such a complex z does not exist</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">such a complex z does not exist</span> </div>

answer:
<div class="solution">Putting \(z=a+2 i\) in the given equation and comparing imaginary parts, we get \(\mathrm{a}^2+4=\mathrm{a}^2\), which is not possible.</div>


title: if-a-0-b-0-c-0-and-a-b-c-are-distinct-then-a-b-b-c-c-a-is-greater-than
slug: if-a-0-b-0-c-0-and-a-b-c-are-distinct-then-a-b-b-c-c-a-is-greater-than-66723
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $a&gt;0, b&gt;0, c&gt;0$ and $a, b, c$ are distinct, then $(a+$b) $(b+c)(c+a)$ is greater than</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2(a+b+c)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3(a+b+c)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$6 \mathrm{abc}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$8 \mathrm{abc}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 \mathrm{abc}$</span> </div>

answer:
<div class="solution">$\mathrm{AM}&gt;\mathrm{GM}$
<br/> <br/>$\Rightarrow \frac{\mathrm{a}+\mathrm{b}}{2}&gt;\sqrt{\mathrm{ab}}, \frac{\mathrm{b}+\mathrm{c}}{2}&gt;\sqrt{\mathrm{bc}}, \frac{\mathrm{c}+\mathrm{a}}{2}&gt;\sqrt{\mathrm{ca}}$
<br/> <br/>So, $(a+b)(b+c)(c+a)&gt;8 a b c$</div>


title: if-a-1-1-1-1-then-a-100
slug: if-a-1-1-1-1-then-a-100-87134
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{A}=\left[\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 1\end{array}\right]$ then $\mathbf{A}^{100}$ :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{100} \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2^{99} \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{101} \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2^{99} \mathrm{~A}$</span> </div>

answer:
<div class="solution">Let $A=\left[\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 1\end{array}\right]$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{A}^{2}=\left[\begin{array}{ll}
<br/> <br/>1 &amp; 1 \\
<br/> <br/>1 &amp; 1
<br/> <br/>\end{array}\right]\left[\begin{array}{ll}
<br/> <br/>1 &amp; 1 \\
<br/> <br/>1 &amp; 1
<br/> <br/>\end{array}\right]=2\left[\begin{array}{ll}
<br/> <br/>1 &amp; 1 \\
<br/> <br/>1 &amp; 1
<br/> <br/>\end{array}\right]=2 \mathrm{~A} \\
<br/> <br/>\mathrm{~A}^{3}=2^{2}\left[\begin{array}{ll}
<br/> <br/>1 &amp; 1 \\
<br/> <br/>1 &amp; 1
<br/> <br/>\end{array}\right], \mathrm{A}^{4}=2^{3}\left[\begin{array}{ll}
<br/> <br/>1 &amp; 1 \\
<br/> <br/>1 &amp; 1
<br/> <br/>\end{array}\right] \\
<br/> <br/>\mathrm{A}^{3}=2^{2} \mathrm{~A}, \quad \mathrm{~A}^{4}=2^{3} \mathrm{~A} \\
<br/> <br/>\therefore \mathrm{A}^{\mathrm{n}}=2^{\mathrm{n}-1}\left[\begin{array}{ll}
<br/> <br/>1 &amp; 1 \\
<br/> <br/>1 &amp; 1
<br/> <br/>\end{array}\right] \\
<br/> <br/>\Rightarrow \mathrm{A}^{100}=2^{100-1} \mathrm{~A} \therefore \mathrm{A}^{100}=2^{99} \mathrm{~A}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: if-a-1-b-1-c-1-are-in-a-p-then-a-1-b-1-c-1-b-1-c-1-a-1-is-equal-to
slug: if-a-1-b-1-c-1-are-in-a-p-then-a-1-b-1-c-1-b-1-c-1-a-1-is-equal-to-59307
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}}$ are in A. P., then $\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}-\frac{1}{\mathrm{c}}\right) \left(\frac{1}{b}+\frac{1}{c}-\frac{1}{a}\right)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{4}{a c}-\frac{3}{b^{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{b^{2}-a c}{a^{2} b^{2} c^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4}{a c}-\frac{1}{b^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{a c}-\frac{3}{b^{2}}$</span> </div>

answer:
<div class="solution">$\frac{1}{\mathrm{a}}-\frac{1}{\mathrm{~b}}=\frac{1}{\mathrm{~b}}-\frac{1}{\mathrm{c}}$
<br/> <br/>$\therefore\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}-\frac{1}{\mathrm{c}}\right)\left(\frac{1}{\mathrm{~b}}+\frac{1}{\mathrm{c}}-\frac{1}{\mathrm{a}}\right)$
<br/> <br/>$=\left(\frac{2}{\mathrm{a}}-\frac{1}{\mathrm{~b}}\right)\left(\frac{2}{\mathrm{c}}-\frac{1}{\mathrm{~b}}\right)=\frac{4}{\mathrm{ac}}-\frac{1}{\mathrm{~b}}\left(\frac{2}{\mathrm{a}}+\frac{2}{\mathrm{c}}\right)+\frac{1}{\mathrm{~b}^{2}}$
<br/> <br/>$=\frac{4}{\mathrm{ac}}-\frac{2}{\mathrm{~b}}\left(\frac{2}{\mathrm{~b}}\right)+\frac{1}{\mathrm{~b}^{2}}=\frac{4}{\mathrm{ac}}-\frac{3}{\mathrm{~b}^{2}}$</div>


title: if-a-2-x-4-b-2-y-4-c-4-then-the-maximum-value-of-xy-is
slug: if-a-2-x-4-b-2-y-4-c-4-then-the-maximum-value-of-xy-is-46918
question: <div class="question">If $a^{2} x^{4}+b^{2} y^{4}=c^{4}$, then the maximum value of xy is</div>
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{c}}{\sqrt{\mathrm{ab}}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{c^{2}}{2 \sqrt{a b}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{c}}{2 \sqrt{\mathrm{ab}}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{c^{2}}{\sqrt{2 a b}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{c^{2}}{\sqrt{2 a b}}$</span> </div>

answer:
<div class="solution">If the sum of two positive quantities is a constant, then their product is maximum, when they are equal. $\therefore \mathrm{a}^{2} \mathrm{x}^{4} \cdot \mathrm{b}^{2} \mathrm{y}^{2}$ is maximum when
<br/> <br/>$$
<br/> <br/>a^{2} x^{4}=b^{2} y^{4}=\frac{1}{2}\left(a^{2} x^{4}=\frac{1}{2}\left(a^{2} x^{4}+b^{2} y^{4}\right)=\frac{c^{4}}{2}\right.
<br/> <br/>$$
<br/> <br/>$\therefore$ maximum value of $a^{2} x^{4} \cdot b^{2} y^{4}=\frac{c^{4}}{2} \cdot \frac{c^{4}}{2}=\frac{c^{8}}{4}$
<br/> <br/>Maximum value of $x y=\left(\frac{c^{8}}{4 a^{2} b^{2}}\right)^{1 / 4}=\frac{c^{2}}{\sqrt{2 a b}}$</div>


title: if-a-and-b-are-2-times-2-matrices-then-which-of-the-following-is-true
slug: if-a-and-b-are-2-times-2-matrices-then-which-of-the-following-is-true-54741
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(A\) and \(B\) are \(2 \times 2\) matrices, then which of the following is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((\mathrm{A}+\mathrm{B})^2=\mathrm{A}^2+\mathrm{B}^2+2 \mathrm{AB}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((A-B)^2=A^2+B^2-2 A B\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\((A-B)(A+B)=A^2+A B-B A-B^2\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\((A-B)(A+B)=A^2-B^2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((A-B)(A+B)=A^2+A B-B A-B^2\)</span> </div>

answer:
<div class="solution">For two \(2 \times 2\) matrices, \(A ~\&amp;~ B\)<br/>\(\begin{aligned}<br/>&amp; (A-B) \times(A+B) \\<br/>&amp; =A \times A+A \times B-B \times A-B \times B \\<br/>&amp; =A^2-B^2+A B-B A<br/>\end{aligned}\)<br/>Hence, \((A-B)(A+B)=A^2+A B-B A-B^2\)</div>


title: if-a-and-b-are-mutually-exclusive-events-and-if-p-b-3-1-p-a-b-21-13-then-p-a-is-equal-to
slug: if-a-and-b-are-mutually-exclusive-events-and-if-p-b-3-1-p-a-b-21-13-then-p-a-is-equal-to-69638
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A$ and $B$ are mutually exclusive events and if $\mathrm{P}(\mathrm{B})=\frac{1}{3}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{13}{21}$, then $\mathrm{P}(\mathrm{A})$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>1/7</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>4/7</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>2/7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>5/7</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>2/7</span> </div>

answer:
<div class="solution">For mutually exclusive events<br/>$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \Rightarrow \mathrm{P}(\mathrm{A})=\frac{2}{7}$</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
if-a-and-b-are-positive-acute-angles-satisfying-3-cos-2-a-2-cos-2-b-4-and-s-i-n-b-3-s-i-n-a-c-o-s-a-2-c-o-s-b-then-the-value-of-a-2-b-is-equal-to

slug:
if-a-and-b-are-positive-acute-angles-satisfying-3-cos-2-a-2-cos-2-b-4-and-s-i-n-b-3-s-i-n-a-c-o-s-a-2-c-o-s-b-then-the-value-of-a-2-b-is-equal-to-89109

question:
<div class="question">If $A$ and $B$ are positive acute angles satisfying $3 \cos ^{2} A+2 \cos ^{2} B=4$ and $\frac{3 \sin A}{\sin B}=\frac{2 \cos B}{\cos A}$
<br/> <br/>Then the value of $\mathrm{A}+2 \mathrm{~B}$ is equal to :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{6}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{2}$</span> </div>

answer:
<div class="solution">Given, $3 \cos ^{2} \mathrm{~A}+2 \cos ^{2} \mathrm{~B}=4$
<br/> <br/>$\Rightarrow 2 \cos ^{2} \mathrm{~B}-1=4-3 \cos ^{2} \mathrm{~A}-1$
<br/> <br/>$\Rightarrow \cos 2 \mathrm{~B}=3\left(1-\cos ^{2} \mathrm{~A}\right)=3 \sin ^{2} \mathrm{~A} \ldots(1)$
<br/> <br/>and $2 \cos \mathrm{B} \sin \mathrm{B}=3 \sin \mathrm{A} \cos \mathrm{A}$
<br/> <br/>$\sin 2 \mathrm{~B}=3 \sin \mathrm{A} \cos \mathrm{A}$
<br/> <br/>Now, $\cos (\mathrm{A}+2 \mathrm{~B})=\cos \mathrm{A} \cos 2 \mathrm{~B}-\sin \mathrm{A}$
<br/> <br/>$\sin 2 \mathrm{~B}$
<br/> <br/>$=\cos A\left(3 \sin ^{2} A\right)-\sin A(3 \sin A \cos A)=0$
<br/> <br/>[using eqs. (1) and (2)] $\Rightarrow A+2 B=\frac{\pi}{2}$</div>


title: if-a-and-bare-nonzero-roots-of-x-2-a-x-b-0-then-the-least-value-of-x-2-a-x-b-is
slug: if-a-and-bare-nonzero-roots-of-x-2-a-x-b-0-then-the-least-value-of-x-2-a-x-b-is-70263
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If a and bare non-zero roots of $x^{2}+a x+b=0$ then the least value of $\mathrm{x}^{2}+a \mathrm{x}+\mathrm{b}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{9}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{9}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{9}{4}$</span> </div>

answer:
<div class="solution">As given a and bare the roots of the equat
<br/> <br/>$x^{2}+a x+b=0$
<br/> <br/>$\Rightarrow$ sum of roots, $a+b=-a$
<br/> <br/>$\Rightarrow \quad \mathrm{b}=-2 \mathrm{a}$
<br/> <br/>and product of roots, $a b=b$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \quad a b-b=0 \\
<br/> <br/>\Rightarrow \quad b(a-1)=0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>if $b=0$ then $a=0$
<br/> <br/>if $b \neq 0$ then $a=1$ and $b=-2$
<br/> <br/>so, the expression will be,
<br/> <br/>$f(x)=x^{2}+x-2$
<br/> <br/>$$
<br/> <br/>=x^{2}+2 \cdot \frac{1}{2} x+\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}-2
<br/> <br/>$$
<br/> <br/>$\Rightarrow \mathrm{f}(\mathrm{x})=\left(\mathrm{x}+\frac{1}{2}\right)^{2}-\frac{9}{4}$
<br/> <br/>So, $\mathrm{f}(\mathrm{x})$ will be minimum, if $\left(\mathrm{x}+\frac{1}{2}\right)^{2}=0$
<br/> <br/>i.e. when $x=-\frac{1}{2}$
<br/> <br/>$\Rightarrow$ minimum value of function $=-\frac{9}{4}$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-a-and-mathrmb-are-mutually-exclusive-events-and-if-mathrmpmathrmbfrac13-mathrmpmathrma-cup-mathrmbfrac1321-then-mathrmpmathrma-is-equal-to

slug:
if-a-and-mathrmb-are-mutually-exclusive-events-and-if-mathrmpmathrmbfrac13-mathrmpmathrma-cup-mathrmbfrac1321-then-mathrmpmathrma-is-equal-to-10010

question:
<div class="question">If \(A\) and \(\mathrm{B}\) are mutually exclusive events and if \(\mathrm{P}(\mathrm{B})=\frac{1}{3}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{13}{21}\), then \(\mathrm{P}(\mathrm{A})\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(1 / 7\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(4 / 7\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(2 / 7\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(5 / 7\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 / 7\)</span> </div>

answer:
<div class="solution">For mutually exclusive events<br/>\(\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \Rightarrow \mathrm{P}(\mathrm{A})=\frac{2}{7}\)</div>


title: if-a-b-a-c-and-a-b-a-c-then-correct-statement-is
slug: if-a-b-a-c-and-a-b-a-c-then-correct-statement-is-58568
question: <div class="question">If $a$.b $=a$.c and $a \times b=a \times c$, then correct statement is</div>
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{a} \|(\mathrm{b}-\mathrm{c})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a \perp(b-c)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$a=0$ or $b=c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a=0$ or $b=c$</span> </div>

answer:
<div class="solution">$a . b .=a \cdot c \Rightarrow a \cdot(b-c)=0$
<br/>$\Rightarrow \mathrm{a}=0 \mathrm{or} \mathrm{b}-\mathrm{c}=0$ or $\mathrm{a} \perp(\mathrm{b}-\mathrm{c})$
<br/>$\Rightarrow \mathrm{a}=0$ or $\mathrm{b}=\mathrm{c}$ or $\mathrm{a} \perp(\mathrm{b}-\mathrm{c})$
<br/>Also a $x b=a x c \Rightarrow a \times(b-c)=0$
<br/>$\Rightarrow \mathrm{a}=0$ or $\mathrm{b}-\mathrm{c}=0$ or a $\|(\mathrm{b}-\mathrm{c})$
<br/>$\Rightarrow a=0$ or $b=c$ or $a \|(b-c)$
<br/>Observing to ( 1 ) and ( 2 ) we find that $\mathrm{a}=0$ or $\mathrm{b}=\mathrm{c}$</div>


title: if-a-b-and-c-are-real-numbers-then-the-roots-of-the-equation-x-a-x-b-x-b-x-c-x-c-x-a-0-are-always-1
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">real</span> </div>
source_db: MarksBatch2_P1.db
slug:
if-a-b-and-c-are-real-numbers-then-the-roots-of-the-equation-x-a-x-b-x-b-x-c-x-c-x-a-0-are-always-1-90717

question:
<div class="question">If $a, b$ and $c$ are real numbers then the roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x$$-a)=0$ are always</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">real</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">imaginary</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">positive</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">negative</span> </li> </ul>

answer:
<div class="solution">Given equation is $(x-a)(x-b)+(x-b)$ $(x-c)+(x-c)(x-a)=0$
<br/> <br/>$\Rightarrow 3 x^{2}-2(b+a+c) x+a b+b c+c a=0$
<br/> <br/>Now, here $A=3, B=-2(a+b+c)$
<br/> <br/>$C=a b+b c+c a$
<br/> <br/>$\therefore D=\sqrt{B^{2}-4 A C}$
<br/> <br/>$=\sqrt{(-2(a+b+c))^{2}-4(3)(a b+b c+c a)}$
<br/> <br/>$=\sqrt{4(a+b+c)^{2}-12(a b+b c+c a)}$
<br/> <br/>$=2 \sqrt{a^{2}+b^{2}+c^{2}-a b-b c-c a}$
<br/> <br/>$=2 \sqrt{\frac{1}{2}\left\{(a-b)^{2}-(b-c)^{2}+(c-a)^{2}\right\}} \geq 0$</div>


title: if-a-b-c-2-then
slug: if-a-b-c-2-then-47731
question: <div class="question">If $\mathrm{A}+\mathrm{B}+\mathrm{C}=\frac{\pi}{2}$ then</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\tan A \tan B+\tan B \tan C+\tan C \tan A=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cot A+\cot B+\cot C=\cot A \cot B \cot C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}=1+4 \sin \mathrm{A} \sin \mathrm{B} \sin C$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">All three are correct</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">All three are correct</span> </div>

answer:
<div class="solution">$\mathrm{B}+\mathrm{C}=\frac{\pi}{2}-\mathrm{A} \Rightarrow \tan (\mathrm{B}+\mathrm{C})=\cot \mathrm{A}$
<br/> <br/>$\Rightarrow \frac{\tan \mathrm{B}+\tan \mathrm{C}}{1-\tan \mathrm{B} \tan \mathrm{C}}=\cot \mathrm{A}$
<br/> <br/>$\Rightarrow \tan \mathrm{A} \tan \mathrm{B}+\tan \mathrm{B} \tan \mathrm{C}+\tan \mathrm{C} \tan \mathrm{A}=1$
<br/> <br/>$\Rightarrow \tan \mathrm{A} \tan \mathrm{B} \tan \mathrm{C}(\cot \mathrm{C}+\cot \mathrm{A}+\cot \mathrm{B})=1$
<br/> <br/>$\Rightarrow \cot \mathrm{A}+\cot \mathrm{B}+\cot \mathrm{C}=\cot \mathrm{A} \cot \mathrm{B} \cot \mathrm{C}$
<br/> <br/>Again $\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}$
<br/> <br/>$=2 \cos (\mathrm{A}+\mathrm{B}) \cos (\mathrm{A}-\mathrm{B})+\cos 2 \mathrm{C}$
<br/> <br/>$=2 \cos \left(\frac{\pi}{2}-C\right) \cos (A-B)+1-2 \sin ^{2} C$
<br/> <br/>$=2 \sin C[\cos (A-B)-\sin C]+1$
<br/> <br/>$=2 \sin \mathrm{C}\left[\cos (\mathrm{A}-\mathrm{B})-\sin \frac{\pi}{2}-(\mathrm{A}+\mathrm{B})\right]+1$
<br/> <br/>$\begin{array}{l}
<br/> <br/>=2 \sin C[\cos (A-B)-\cos (A+B)]+1 \\
<br/> <br/>=4 \sin A \sin B \sin C+1
<br/> <br/>\end{array}$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-in-gp-and-log-a-log-2-b-log-2-b-log-3-c-and-log-3-c-log-a-are-in-ap-then-a-b-and-c-are-the-lengths-of-the-sides-of-a-triangle-which-is

slug:
if-a-b-c-are-in-gp-and-log-a-log-2-b-log-2-b-log-3-c-and-log-3-c-log-a-are-in-ap-then-a-b-and-c-are-the-lengths-of-the-sides-of-a-triangle-which-is-38563

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>GP</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mi>a</mi><mo>-</mo><mi>log</mi><mn>2</mn><mi>b</mi><mo>,</mo><mo> </mo><mi>log</mi><mn>2</mn><mi>b</mi><mo>-</mo><mi>log</mi><mn>3</mn><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mn>3</mn><mi>c</mi><mo>-</mo><mi>log</mi><mi>a</mi></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AP</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> are the lengths of the sides of a triangle, which is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">equilateral</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">right angled</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">acute-angled</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">obtuse angled</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">obtuse angled</span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>GP</mi></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><mi>c</mi></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mi>log</mi><mn>2</mn><mi>b</mi><mo>-</mo><mi>log</mi><mn>3</mn><mi>c</mi></mrow></mfenced><mo>=</mo><mi>log</mi><mi>a</mi><mo>-</mo><mi>log</mi><mn>2</mn><mi>b</mi><mo>+</mo><mi>log</mi><mn>3</mn><mi>c</mi><mo>-</mo><mi>log</mi><mi>a</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>b</mi><mo>=</mo><mn>3</mn><mi>c</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mi>a</mi></mrow><mn>3</mn></mfrac><mo>&gt;</mo><mi>c</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mfrac><mrow><mn>10</mn><mi>a</mi></mrow><mn>9</mn></mfrac><mo>&gt;</mo><mi>a</mi></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>+</mo><mi>a</mi><mo>=</mo><mfrac><mrow><mn>13</mn><mi>a</mi></mrow><mn>9</mn></mfrac><mo>&gt;</mo><mi>b</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi></math> are the sides of a triangle.</p><p>Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is the greatest side</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>cos</mi><mi>A</mi><mo>=</mo><mfrac><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>b</mi><mi>c</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>29</mn><mn>48</mn></mfrac><mo>&lt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>∆</mo><mi>ABC</mi></math> is an obtuse angled triangle.</p><p>Hence, option (d) is correct.</p></div>


tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-noncoplaner-vectors-such-that-b-c-a-c-a-b-a-b-c-then-which-of-the-following-is-not-true

slug:
if-a-b-c-are-noncoplaner-vectors-such-that-b-c-a-c-a-b-a-b-c-then-which-of-the-following-is-not-true-40854

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>→</mo></mover><mo>,</mo><mover><mi>b</mi><mo>→</mo></mover><mo>,</mo><mover><mi>c</mi><mo>→</mo></mover></math> are non-coplaner vectors such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">×</mo><mover><mi mathvariant="italic">c</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">=</mo><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">;</mo><mo mathvariant="italic"> </mo><mover><mi mathvariant="italic">c</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">×</mo><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">=</mo><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">;</mo><mo mathvariant="italic"> </mo><mover><mi mathvariant="italic">a</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">×</mo><mover><mi mathvariant="italic">b</mi><mo mathvariant="italic">→</mo></mover><mo mathvariant="italic">=</mo><mover><mi mathvariant="italic">c</mi><mo mathvariant="italic">→</mo></mover></math>, then which of the following is not TRUE?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>-</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mo>|</mo><mover><mi>c</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mtable><mtr><mtd><mover><mi>a</mi><mo>→</mo></mover></mtd><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd><mtd><mover><mi>c</mi><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo><mo>|</mo><mover><mi>c</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mo>|</mo><mover><mi>c</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Given, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>→</mo></mover><mo>×</mo><mover><mi>b</mi><mo>→</mo></mover><mo>=</mo><mover><mi>c</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover><mi>c</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>c</mi><mo>→</mo></mover><mo>×</mo><mover><mi>a</mi><mo>→</mo></mover><mo>=</mo><mover><mi>b</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mover><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover><mi>c</mi><mo>→</mo></mover><mo>)</mo><mo>·</mo><mover><mi>a</mi><mo>→</mo></mover><mo>=</mo><mover><mi>a</mi><mo>→</mo></mover><mo>·</mo><mover><mi>a</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mover><mi>c</mi><mo>→</mo></mover><mo>×</mo><mover><mi>a</mi><mo>→</mo></mover><mo>)</mo><mo>·</mo><mover><mi>b</mi><mo>→</mo></mover><mo>=</mo><mover><mi>b</mi><mo>→</mo></mover><mo>·</mo><mover><mi>b</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mover><mi>a</mi><mo>→</mo></mover><mn>2</mn></msup><mo>-</mo><msup><mover><mi>b</mi><mo>→</mo></mover><mn>2</mn></msup><mo>=</mo><mo>(</mo><mover><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover><mi>c</mi><mo>→</mo></mover><mo>)</mo><mo>·</mo><mover><mi>a</mi><mo>→</mo></mover><mo>-</mo><mo>(</mo><mover><mi>c</mi><mo>→</mo></mover><mo>×</mo><mover><mi>a</mi><mo>→</mo></mover><mo>)</mo><mo>·</mo><mover><mi>b</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>[</mo><mtable><mtr><mtd><mover><mi>c</mi><mo>→</mo></mover></mtd><mtd><mover><mi>a</mi><mo>→</mo></mover></mtd><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd></mtr></mtable><mo>]</mo><mo>-</mo><mo>[</mo><mtable><mtr><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd><mtd><mover><mi>c</mi><mo>→</mo></mover></mtd><mtd><mover><mi>a</mi><mo>→</mo></mover></mtd></mtr></mtable><mo>]</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><msup><mo>|</mo><mn>2</mn></msup><mo>-</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><msup><mo>|</mo><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>-</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo></math></p><p>We know that, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mover><mi>a</mi><mo>→</mo></mover><mo>×</mo><mover><mi>b</mi><mo>→</mo></mover><mo> </mo><mo> </mo><mover><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover><mi>c</mi><mo>→</mo></mover><mo> </mo><mo> </mo><mover><mi>c</mi><mo>→</mo></mover><mo>×</mo><mover><mi>a</mi><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mo>[</mo><mover><mi>a</mi><mo>→</mo></mover><mover><mi>b</mi><mo>→</mo></mover><mover><mi>c</mi><mo>→</mo></mover><msup><mo>]</mo><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><mover><mi>a</mi><mo>→</mo></mover></mtd><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd><mtd><mover><mi>c</mi><mo>→</mo></mover></mtd></mtr></mtable></mtd></mtr></mtable></mfenced><mo>=</mo><msup><mfenced close="]" open="["><mtable columnalign="left"><mtr><mtd><mover><mi>a</mi><mo>→</mo></mover></mtd><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd><mtd><mover><mi>c</mi><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced close="]" open="["><mtable><mtr><mtd><mover><mi>a</mi><mo>→</mo></mover></mtd><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd><mtd><mover><mi>c</mi><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo><mo>|</mo><mover><mi>c</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>|</mo><mover><mi>a</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mo>|</mo><mover><mi>b</mi><mo>→</mo></mover><mo>|</mo><mo>=</mo><mo>|</mo><mover><mi>c</mi><mo>→</mo></mover><mo>|</mo><mo>≠</mo><mn>2</mn></math></p></div>


title: if-a-b-c-are-positive-numbers-then-least-value-of-a-b-c-a-1-b-1-c-1-is
slug: if-a-b-c-are-positive-numbers-then-least-value-of-a-b-c-a-1-b-1-c-1-is-20399
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $a, b, c$ are positive numbers, then least value of $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

answer:
<div class="solution">$\mathrm{AM} \geq \mathrm{HM}$
<br/> <br/>$\Rightarrow \frac{a+b+c}{3} \geq \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$
<br/> <br/>$\Rightarrow(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 9$</div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">13</span> </div>
source_db: MarksBatch2_P1.db
title:
if-a-b-c-are-three-natural-numbers-in-ap-and-a-b-c-21-then-the-possible-number-of-values-of-the-ordered-triplet-a-b-c-is

slug:
if-a-b-c-are-three-natural-numbers-in-ap-and-a-b-c-21-then-the-possible-number-of-values-of-the-ordered-triplet-a-b-c-is-83134

question:
<div class="question">If a, b, c are three natural numbers in AP and a + $\mathrm{b}+\mathrm{c}=21$ then the possible number of values of the ordered triplet $(\mathrm{a}, \mathrm{b}, \mathrm{c})$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">15</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">14</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">13</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$\quad$ Let $a=b-d$ and $c=b+d$, then $a+b+c=21$ $\Rightarrow \mathrm{b}=7$
<br/> <br/>So, the equation is $\mathrm{a}+\mathrm{c}=14$
<br/> <br/>$\therefore$ No. of solution $=$ coeff. of $x^{14}$ in $\left(x+x^{2}+\ldots .\right)$ $={ }^{13} \mathrm{C}_{12}=13$</div>


title: if-a-b-c-are-three-unit-vectors-such-that-a-b-c-0-where-0-is-null-vector-then-a-b-b-c-c-a-is
slug: if-a-b-c-are-three-unit-vectors-such-that-a-b-c-0-where-0-is-null-vector-then-a-b-b-c-c-a-is-39911
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}, \vec{b}, \vec{c}$ are three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},$ where $\overrightarrow{0}$ is null vector, then $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\frac{3}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{3}{2}$</span> </div>

answer:
<div class="solution">We have $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$
<br/> <br/>$|\vec{a}+\vec{b}+\vec{c}|=0 \Rightarrow|\vec{a}+\vec{b}+\vec{c}|^{2}=0$
<br/> <br/>$\Rightarrow|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}$
<br/> <br/>$+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$
<br/> <br/>$\Rightarrow 1+1+1+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$
<br/> <br/>$\Rightarrow \vec{a} \cdot \vec{b}+\bar{b} \cdot \vec{c}+\bar{c} \cdot \bar{a}=-\frac{3}{2}$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
title:
if-a-c-b-are-in-gp-then-the-area-of-the-triangle-formed-by-the-lines-a-x-b-y-c-0-with-the-coordinates-axes-is-equal-to

slug:
if-a-c-b-are-in-gp-then-the-area-of-the-triangle-formed-by-the-lines-a-x-b-y-c-0-with-the-coordinates-axes-is-equal-to-30744

question:
<div class="question">If $a, c, b$ are in G.P., then the area of the triangle formed by the lines $a x+b y+c=0$ with the coordinates axes is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Area of the triangle $=\frac{1}{2}(\mathrm{x}$ intercept $) \times(\mathrm{y}$ intercept $)$
<br/> <br/>$=\frac{1}{2}\left(-\frac{\mathrm{c}}{\mathrm{a}}\right)\left(-\frac{\mathrm{c}}{\mathrm{b}}\right)=\frac{1}{2} \frac{\mathrm{c}^{2}}{\mathrm{ab}}=\frac{1}{2} \mathrm{unit}$
<br/> <br/>$\left[\because \mathrm{a}, \mathrm{c}, \mathrm{b}\right.$ are in $\left.\mathrm{G} . \mathrm{P} . \Rightarrow \mathrm{c}^{2}=\mathrm{ab}\right]$</div>


title: if-a-chord-of-the-circle-x-2-y-2-8-makes-equal-intercets-of-length-a-on-the-coordinate-axes-then
tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
slug:
if-a-chord-of-the-circle-x-2-y-2-8-makes-equal-intercets-of-length-a-on-the-coordinate-axes-then-59025

question:
<div class="question">If a chord of the circle $x^{2}+y^{2}=8$ makes equal intercets of length a on the coordinate axes, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$|\mathrm{a}|&lt;8$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$|\mathrm{a}|&lt;4 \sqrt{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$|\mathrm{a}|&lt;4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|\mathrm{a}|&gt;4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$|\mathrm{a}|&lt;4$</span> </div>

answer:
<div class="solution">Since the chord makes equal intercepts of length a on the coordinate axes. So, its equation can be written as $\mathrm{x} \pm \mathrm{y}=\pm \mathrm{a}$. This line meets the given circle at two distinct points. So, length of the perpendicular from the centre (0,0) of the given circle must be less than the radius.
<br/> <br/>i.e. $\left|\pm \frac{\mathrm{a}}{\sqrt{2}}\right|&lt;\sqrt{8} \Rightarrow \mathrm{a}^{2}&lt;16 \Rightarrow|\mathrm{a}|&lt;4$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
if-a-circular-plate-is-heated-uniformly-its-area-expands-3-c-times-as-fast-as-its-radius-then-the-value-of-c-when-the-radius-is-6-units-is

slug:
if-a-circular-plate-is-heated-uniformly-its-area-expands-3-c-times-as-fast-as-its-radius-then-the-value-of-c-when-the-radius-is-6-units-is-55491

question:
<div class="question">If a circular plate is heated uniformly, its area expands \(3 c\) times as fast as its radius, then the value of \(c\) when the radius is 6 units, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(4 \pi\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(2 \pi\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(6 \pi\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(3 \pi\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(4 \pi\)</span> </div>

answer:
<div class="solution">Let \(A\) sq. units in the area measure when the radius is \(r\) units.<br/>their \(A=\pi r^2\)<br/>Differentiate both side w.r.t ' \(t\) '<br/>\(\frac{d A}{d t}=2 \pi r \frac{d r}{d t}\) ...(i))<br/>We have, \(\frac{d A}{d t}=3 c \frac{d r}{d t}\)<br/>From eqn (i), we get<br/>\(3 c \cdot \frac{d r}{d t}=2 \pi r \cdot \frac{d r}{d t} \Rightarrow 3 c=2 \pi r\)<br/>Now, \(c=\frac{2}{3} \pi(6)=4 \pi\) when \(r=6\)</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-a-drop-of-liquid-breaks-into-smaller-droplets-it-results-in-lowering-of-temperature-of-the-droplets-let-a-drop-of-radius-r-break-into-n-small-dropl-1

slug:
if-a-drop-of-liquid-breaks-into-smaller-droplets-it-results-in-lowering-of-temperature-of-the-droplets-let-a-drop-of-radius-r-break-into-n-small-dropl-1-61512

question:
<div class="question">If a drop of liquid breaks into smaller droplets, it results in lowering of temperature of the droplets. Let a drop of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>, break into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> small droplets each of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, then decrease (drop) in temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> (given, specific heat of liquid drop<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>S</mi></math> and surface tension <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>T</mi></math> )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><mi>ρ</mi><mi>S</mi></mrow></mfrac><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mi>r</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>R</mi></mfrac></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mn>2</mn><mi>T</mi></mrow><mrow><mi>ρ</mi><mi>S</mi></mrow></mfrac><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mi>r</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>R</mi></mfrac></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>R</mi></mrow><mrow><mi>ρ</mi><mi>S</mi></mrow></mfrac><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>r</mi></mfrac></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><mi>ρ</mi><mi>S</mi></mrow></mfrac><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>r</mi></mfrac></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><mi>ρ</mi><mi>S</mi></mrow></mfrac><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>r</mi></mfrac></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>since, volume remains unchanged, during this phenomenon, so</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>R</mi><mn>3</mn></msup><mo>=</mo><mi>N</mi><mo>×</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>3</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><msup><mi>R</mi><mn>3</mn></msup><msup><mi>r</mi><mn>3</mn></msup></mfrac></math></p><p>Now, change in surface area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mi>π</mi><msup><mi>R</mi><mn>2</mn></msup><mo>-</mo><mi>N</mi><mn>4</mn><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mi>π</mi><mfenced><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>-</mo><mi>N</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfenced></math></p><p>Energy released <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>Δ</mi><mi>U</mi><mo>)</mo><mo>=</mo><mi>T</mi><mo>×</mo></math> change in surface area</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>T</mi><mo>×</mo><mn>4</mn><mi>π</mi><mfenced close="]" open="["><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>-</mo><mi>N</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfenced></math></p><p>Here, all this energy released is al the cost of lowering the temperature and mass of the big drop of liquid <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>R</mi><mn>2</mn></msup><mi>ρ</mi></math></p><p>Now, change in temperature, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Δ</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>Δ</mi><mi>U</mi></mrow><mrow><mi>m</mi><mi>s</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>T</mi><mo>×</mo><mn>4</mn><mi>π</mi><mfenced><mrow><msup><mi>R</mi><mn>2</mn></msup><mo>-</mo><mi>N</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfenced></mrow><mrow><mfenced><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>R</mi><mn>3</mn></msup><mi>ρ</mi></mrow></mfenced><mi>S</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><mi>μ</mi><mi>S</mi></mrow></mfrac><mfenced><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><mo>-</mo><mfrac><mrow><mi>N</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><msup><mi>R</mi><mn>3</mn></msup></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><mi>ρ</mi><mi>S</mi></mrow></mfrac><mfenced><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><mo>-</mo><mfrac><mrow><msup><mi>R</mi><mn>3</mn></msup><mo>×</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mrow><msup><mi>r</mi><mn>3</mn></msup><mo>×</mo><msup><mi>R</mi><mn>3</mn></msup></mrow></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><mi>ρ</mi><mi>S</mi></mrow></mfrac><mfenced><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>r</mi></mfrac></mrow></mfenced></math></p></div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
if-a-flagstaff-of-6-metres-high-placed-on-the-top-of-a-tower-throws-a-shadow-of-2-sqrt3-metres-along-the-ground-then-the-angle-in-degrees-that-the-sun

slug:
if-a-flagstaff-of-6-metres-high-placed-on-the-top-of-a-tower-throws-a-shadow-of-2-sqrt3-metres-along-the-ground-then-the-angle-in-degrees-that-the-sun-60578

question:
<div class="question">If a flagstaff of 6 metres high placed on the top of a tower throws a shadow of \(2 \sqrt{3}\) metres along the ground, then the angle (in degrees) that the sun makes with the ground is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(60^{\circ}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(80^{\circ}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(75^{\circ}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(60^{\circ}\)</span> </div>

answer:
<div class="solution">Accordingly,<br/>\(\tan \theta=\frac{h}{x}=\frac{h+6}{x+2 \sqrt{3}}=\frac{6}{2 \sqrt{3}} \Rightarrow \theta=60^{\circ}\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/tQkcpar07B9fqdYJWQ12ih7PgndRF2IP_v_OdhX8gvc.original.fullsize.png"/><br/><br/>[Since the triangles \(A E C\) and \(B D C\) are similar] \(h \cot \alpha=(h-100) \cot \beta\)<br/>\(\therefore h=\frac{100 \cot \beta}{\cot \beta-\cot \alpha}\)</div>


tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
title:
if-a-function-f-r-r-satisfy-the-equation-f-x-y-f-x-f-y-x-y-and-the-function-f-x-is-continuous-at-x-0-then

slug:
if-a-function-f-r-r-satisfy-the-equation-f-x-y-f-x-f-y-x-y-and-the-function-f-x-is-continuous-at-x-0-then-71711

question:
<div class="question">If a function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math> satisfy the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>,</mo><mo>∀</mo><mi>x</mi><mo>,</mo><mi>y</mi></math> and the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is continuous for all positive real values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is continuous for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is continuous for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced></math></p><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> be any point</p><p>Now, at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>a</mi><mo>,</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></munder><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>h</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced><mi>a</mi></mfenced><mo>+</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced><mi>h</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>f</mi><mfenced><mi>a</mi></mfenced><mo>+</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>a</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>a</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is any arbitrary point.</p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is continuous for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p></div>


title: if-a-function-f-x-is-given-by-f-x-1-x-x-x-1-2-x-1-x-2-x-1-3-x-1-x-then-at-x-0-f-x
slug: if-a-function-f-x-is-given-by-f-x-1-x-x-x-1-2-x-1-x-2-x-1-3-x-1-x-then-at-x-0-f-x-15616
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If a function $f(x)$ is given by
<br/> <br/>$f(x)=\frac{x}{1+x}+\frac{x}{(x+1)(2 x+1)}$
<br/> <br/>$+\frac{x}{(2 x+1)(3 x+1)}+\ldots, \infty$, then at $x=0, f(x)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">has no limit</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">is not continuous</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">is continuous but not differentiable</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">is differentiable</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">is not continuous</span> </div>

answer:
<div class="solution">Let
<br/> <br/>$f(x)=\frac{x}{1+x}+\frac{x}{(x+1)(2 x+1)}+\frac{x}{(2 x+1)(3 x+1)}+\ldots \infty$
<br/> <br/>$=\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{x}{[(r-1) x+1](r x+1)}$
<br/> <br/>$=\lim _{x \rightarrow \infty} \sum_{\mathrm{r}=1}^{\mathrm{n}}\left[\frac{\mathrm{x}}{[(\mathrm{r}-1) \mathrm{x}+1]}-\frac{1}{\mathrm{rx}+1}\right]$
<br/> <br/>$=\lim _{n \rightarrow \infty}\left[1-\frac{1}{n x+1}\right]=1$
<br/> <br/>For $x=0,$ we have $f(x)=0$
<br/> <br/>Thus, we have $f(x)=\left\{\begin{array}{ll}1, &amp; x \neq 0 \\ 0, &amp; x=0\end{array}\right.$
<br/> <br/>Clearly, $\lim _{x \rightarrow 0^{-}} \mathrm{f}(\mathrm{x})=\lim _{\mathrm{x} \rightarrow 0^{+}} \mathrm{f}(\mathrm{x}) \neq \mathrm{f}(0)$
<br/> <br/>So, $\mathrm{f}(\mathrm{x})$ is not continuous at $\mathrm{x}=0$.</div>


title: if-a-i-2-j-3-k-b-i-2-j-k-c-3-i-j-and-a-p-b-is-normal-to-c-then-p-is-equal-to
slug: if-a-i-2-j-3-k-b-i-2-j-k-c-3-i-j-and-a-p-b-is-normal-to-c-then-p-is-equal-to-86525
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">a</mi><mo>→</mo></mover><mo>=</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover><mo>,</mo><mo> </mo><mover><mi mathvariant="normal">b</mi><mo>→</mo></mover><mo>=</mo><mo>-</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover><mo>,</mo><mo> </mo><mover><mi mathvariant="normal">c</mi><mo>→</mo></mover><mo>=</mo><mn>3</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">a</mi><mo>→</mo></mover><mo>+</mo><mi>p</mi><mover><mi mathvariant="normal">b</mi><mo>→</mo></mover></math> is normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">c</mi><mo>→</mo></mover></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mover><mi mathvariant="normal">a</mi><mo>→</mo></mover><mo>+</mo><mi>p</mi><mover><mi mathvariant="normal">b</mi><mo>→</mo></mover></mrow></mfenced></math> is normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">c</mi><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfenced><mrow><mover><mi mathvariant="normal">a</mi><mo>→</mo></mover><mo>+</mo><mi>p</mi><mover><mi mathvariant="normal">b</mi><mo>→</mo></mover></mrow></mfenced><mo>·</mo><mover><mi mathvariant="normal">c</mi><mo>→</mo></mover><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mfenced><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><mn>2</mn><mo>+</mo><mn>2</mn><mi>p</mi></mrow></mfenced><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mfenced><mrow><mn>3</mn><mo>+</mo><mi>p</mi></mrow></mfenced><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>·</mo><mfenced><mrow><mn>3</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mn>3</mn><mi>p</mi><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn><mi>p</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>p</mi><mo>=</mo><mn>5</mn></math></p></div>


title: if-a-i-j-k-a-b-1-and-a-b-j-k-then-b-is-1
slug: if-a-i-j-k-a-b-1-and-a-b-j-k-then-b-is-1-53130
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{a} \cdot \vec{b}=1$ and $\vec{a} \times \vec{b}=\vec{j}-\vec{k}$, then $\vec{b}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\hat{i}-\hat{j}+\hat{k}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \hat{j}-\hat{k}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \hat{i}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\hat{i}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\hat{i}$</span> </div>

answer:
<div class="solution">Given $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{a} \cdot \vec{b}=1$ and $\vec{a} \times \vec{b}=\vec{j}-\vec{k}$,<br/>Let $\bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \widehat{k}$<br/>Now, $\hat{j}-\widehat{k}=\bar{a} \times \bar{b}=\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \widehat{k} \\ 1 &amp; 1 &amp; 1 \\ b_2 &amp; b_2 &amp; b_3\end{array}\right|$<br/>$\Rightarrow b_3-b_2=0, b_1-b_3=1, b_2-b_1=-1$<br/>$\Rightarrow b_3=b_2, b_1=b_2+1$<br/>Also, $\bar{a} \cdot \bar{b}=1$<br/>$\Rightarrow b_1+b_2+b_3=1$<br/>$\Rightarrow 3 b_2+1=1$<br/>$\Rightarrow b_2=0$<br/>$\Rightarrow b_1=1, b_3=0$<br/>Thus, $\bar{b}=\hat{i}$</div>


tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
title:
if-a-is-a-vector-of-magnitude-50-collinear-with-the-vector-b-6-i-8-j-2-15-k-and-makes-an-acute-angle-with-the-positive-direction-of-z-axis-then-a-is-e

slug:
if-a-is-a-vector-of-magnitude-50-collinear-with-the-vector-b-6-i-8-j-2-15-k-and-makes-an-acute-angle-with-the-positive-direction-of-z-axis-then-a-is-e-94354

question:
<div class="question">If $\vec{a}$ is a vector of magnitude 50 , collinear with the vector $\vec{b}=6 \hat{i}-8 \hat{j}-\frac{15}{2} \hat{k}$ and makes an acute angle with the positive direction of $Z$ - axis, then $\vec{a}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-24 \hat{i}+32 \hat{j}+30 \hat{k}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$24 \hat{i}-32 \hat{j}-30 \hat{k}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-12 \hat{i}+16 \hat{j}-15 \hat{k}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$12 \hat{i}-16 \hat{j}-15 \hat{k}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-24 \hat{i}+32 \hat{j}+30 \hat{k}$</span> </div>

answer:
<div class="solution">Since $\vec{a}=m \vec{b}$ for some scalar $m$ i.e.,
<br/> <br/>$\begin{array}{l}
<br/> <br/>\vec{a}=m\left(6 \hat{i}-8 \hat{j}-\frac{15}{2} \hat{k}\right) \\
<br/> <br/>\Rightarrow|a|=|m| \sqrt{36+64+\frac{225}{4}} \\
<br/> <br/>\Rightarrow 50=\frac{25}{2}|m| \\
<br/> <br/>\Rightarrow|m|=4 \\
<br/> <br/>\Rightarrow m=\pm 4
<br/> <br/>\end{array}$
<br/> <br/>Since, a makes an acute angle with the positive direction of $Z$ - axis, so its $z$ component must be positive and hence, $m$ must be $-4$.
<br/> <br/>$\therefore a=-4\left(6 \hat{i}+8 \hat{j}-\frac{15}{2} \hat{k}\right)=-24 \hat{i}+32 \hat{j}+30 \hat{k}$</div>


title: if-a-is-an-3-3-non-singular-matrix-such-that-a-a-a-a-and-b-a-1-a-then-b-b-equals-to
slug: if-a-is-an-3-3-non-singular-matrix-such-that-a-a-a-a-and-b-a-1-a-then-b-b-equals-to-48185
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>3</mn></math> non- singular matrix, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><msup><mi>A</mi><mo>'</mo></msup><mo>=</mo><msup><mi>A</mi><mo>'</mo></msup><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mo>'</mo></msup></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><msup><mi>B</mi><mo>'</mo></msup></math> equals to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><msup><mi>B</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo>'</mo></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>+</mo><mi>B</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>B</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math></span> </div>

answer:
<div class="solution"><p>Given,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><msup><mi>A</mi><mo>'</mo></msup><mo>=</mo><msup><mi>A</mi><mo>'</mo></msup><mi>A</mi></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mo>'</mo></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><msup><mi>B</mi><mo>'</mo></msup><mo>=</mo><mfenced><mrow><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mo>'</mo></msup></mrow></mfenced><msup><mfenced><mrow><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mo>'</mo></msup></mrow></mfenced><mo>'</mo></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mo>'</mo></msup></mrow></mfenced><mfenced close="]" open="["><mrow><msup><mfenced><msup><mi>A</mi><mo>'</mo></msup></mfenced><mo>'</mo></msup><msup><mfenced><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo>'</mo></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mo>'</mo></msup></mrow></mfenced><mfenced close="]" open="["><mrow><mi>A</mi><msup><mfenced><msup><mi>A</mi><mo>'</mo></msup></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mo> </mo><msup><mfenced><msup><mi>A</mi><mo>'</mo></msup></mfenced><mo>'</mo></msup><mo>=</mo><mi>A</mi><mo>,</mo><msup><mfenced><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo>'</mo></msup><mo>=</mo><msup><mfenced><msup><mi>A</mi><mo>'</mo></msup></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mi>A</mi><mo>'</mo></msup><mi>A</mi></mrow></mfenced><msup><mfenced><msup><mi>A</mi><mo>'</mo></msup></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced close=""><mrow><mi>A</mi><msup><mi>A</mi><mo>'</mo></msup><msup><mfenced><msup><mi>A</mi><mo>'</mo></msup></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>A</mi></mrow></mfenced><msup><mfenced><mrow><msup><mi>A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>A</mi></mrow></mfenced><mo>'</mo></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>I</mi><mo>·</mo><mi>I</mi><mo>=</mo><mi>I</mi></math></p></div>


title: if-a-log-2-3-b-log-2-5-and-c-log-7-2-then-log-140-63-in-terms-of-a-b-c-is
slug: if-a-log-2-3-b-log-2-5-and-c-log-7-2-then-log-140-63-in-terms-of-a-b-c-is-56848
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mn>3</mn><mo>,</mo><mi>b</mi><mo>=</mo><msub><mi>log</mi><mn>2</mn></msub><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><msub><mi>log</mi><mn>7</mn></msub><mn>2</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>140</mn></msub><mn>63</mn></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>a</mi><mi>c</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>c</mi><mo>+</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>a</mi><mi>c</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>a</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>a</mi><mi>c</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>c</mi><mo>+</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>a</mi></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution"><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>140</mn></msub><mn>63</mn><mo>=</mo><msub><mi>log</mi><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>×</mo><mn>5</mn><mo>×</mo><mn>7</mn></mrow></msub><mo>(</mo><mn>3</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>7</mn><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>log</mi><mn>2</mn></msub><mo>(</mo><mn>3</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>7</mn><mo>)</mo></mrow><mrow><msub><mi>log</mi><mn>2</mn></msub><mfenced><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>×</mo><mn>5</mn><mo>×</mo><mn>7</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>log</mi><mn>2</mn></msub><mn>3</mn><mo>+</mo><msub><mi>log</mi><mn>2</mn></msub><mn>3</mn><mo>+</mo><msub><mi>log</mi><mn>2</mn></msub><mn>7</mn></mrow><mrow><mn>2</mn><msub><mi>log</mi><mn>2</mn></msub><mn>2</mn><mo>+</mo><msub><mi>log</mi><mn>2</mn></msub><mn>5</mn><mo>+</mo><msub><mi>log</mi><mn>2</mn></msub><mn>7</mn></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mfrac><mn>1</mn><mi>c</mi></mfrac></mrow><mrow><mn>2</mn><mo>+</mo><mi>b</mi><mo>+</mo><mfrac><mn>1</mn><mi>c</mi></mfrac></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>a</mi><mi>c</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></p></div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-a-parallelogram-is-cut-by-two-sets-of-m-lines-parallel-to-its-sides-then-the-number-of-parallelogram-thus-formed-is

slug:
if-a-parallelogram-is-cut-by-two-sets-of-m-lines-parallel-to-its-sides-then-the-number-of-parallelogram-thus-formed-is-44581

question:
<div class="question">If a parallelogram is cut by two sets of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> lines parallel to its sides, then the number of parallelogram thus formed, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mi>m</mi></mmultiscripts><mo>×</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mi>m</mi></mmultiscripts></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mmultiscripts></mfenced></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mmultiscripts></mfenced><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mmultiscripts></mfenced><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><p>The two sets of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> parallel lines along with two sets of two parallel lines of the given parallelogram will form two sets of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mfenced></math> parallel lines. Each parallelogram is formed by choosing two parallel lines from each of the above</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Total number of parallelograms</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mmultiscripts><mo>×</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mmultiscripts><mo>=</mo><msup><mfenced><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mmultiscripts></mfenced><mn>2</mn></msup></math></p><p>Hence, option (c) is correct.</p></div>


title: if-a-x-x-is-a-multiple-of-4-and-b-x-x-is-a-multiple-of-6-then-a-b-consists-of-multiples-of
slug: if-a-x-x-is-a-multiple-of-4-and-b-x-x-is-a-multiple-of-6-then-a-b-consists-of-multiples-of-23751
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $A=\{x: x$ is a multiple of 4$\}$ and, $B=\{x: x$ is a multiple of 6$\}$, then $A \cap B$ consists of multiples of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>16
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>12
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>88
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>12
<br/></span> </div>

answer:
<div class="solution">Let $x \in A \cap B$. Then,
<br/>$x \in A$ and $x \in B$
<br/>$\Rightarrow x$ is a multiple of 4 and $x$ is a multiple of 6
<br/>$\Rightarrow x$ is a multiple of 4 and 6 both
<br/>$\therefore x$ is a multiple of 12</div>


title: if-a12345-then-the-number-of-proper-subsets-of-a-is
slug: if-a12345-then-the-number-of-proper-subsets-of-a-is-91597
question: <div class="question">If \(A=\{1,2,3,4,5\}\), then the number of proper subsets of \(A\) is</div>
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">31</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">31</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">38</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">48</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">54</span> </li> </ul>

answer:
<div class="solution">Number of proper subsets of \(A=2^n-1\) Given: \(A=\{1,2,3,4,5\}\)<br/>Here \(n=5\)<br/>\(\therefore\) no. of proper subsets \(=2^5-1\)</div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
if-aleftbeginarrayll1-3-3-2-2-5endarrayright-and-bleftbeginarraycc1-2-0-5-3-1endarrayright-and-mathrmamathrmbmathrmd0-zero-matrix-then-mathrmd-matrix-

slug:
if-aleftbeginarrayll1-3-3-2-2-5endarrayright-and-bleftbeginarraycc1-2-0-5-3-1endarrayright-and-mathrmamathrmbmathrmd0-zero-matrix-then-mathrmd-matrix-93655

question:
<div class="question">If \(A=\left[\begin{array}{ll}1 &amp; 3 \\ 3 &amp; 2 \\ 2 &amp; 5\end{array}\right]\) and \(B=\left[\begin{array}{cc}-1 &amp; -2 \\ 0 &amp; 5 \\ 3 &amp; 1\end{array}\right]\) and \(\mathrm{A}+\mathrm{B}-\mathrm{D}=0\) (zero matrix), then \(\mathrm{D}\) matrix will be -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\left[\begin{array}{ll}0 &amp; 2 \\ 3 &amp; 7 \\ 6 &amp; 5\end{array}\right]\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\left[\begin{array}{ll}0 &amp; 2 \\ 3 &amp; 7 \\ 5 &amp; 6\end{array}\right]\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\left[\begin{array}{ll}0 &amp; 1 \\ 3 &amp; 7 \\ 5 &amp; 6\end{array}\right]\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\left[\begin{array}{cc}0 &amp; -2 \\ -3 &amp; -7 \\ -5 &amp; -6\end{array}\right]\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left[\begin{array}{ll}0 &amp; 1 \\ 3 &amp; 7 \\ 5 &amp; 6\end{array}\right]\)</span> </div>

answer:
<div class="solution">Let \(D=\left[\begin{array}{ll}a &amp; b \\ c &amp; d \\ e &amp; f\end{array}\right]\)<br/>\(\begin{aligned} &amp; \therefore A+B-C=\left[\begin{array}{ll}1 &amp; 3 \\ 3 &amp; 2 \\ 2 &amp; 5\end{array}\right]+\left[\begin{array}{cc}-1 &amp; -2 \\ 0 &amp; 5 \\ 3 &amp; 1\end{array}\right]-\left[\begin{array}{ll}\mathrm{a} &amp; \mathrm{b} \\ \mathrm{c} &amp; \mathrm{d} \\ \mathrm{e} &amp; \mathrm{f}\end{array}\right] \\ &amp; \Rightarrow\left[\begin{array}{ll}1-1-\mathrm{a} &amp; 3-2-\mathrm{b} \\ 3+0-\mathrm{c} &amp; 2+5-\mathrm{d} \\ 2+3-\mathrm{e} &amp; 5+1-\mathrm{f}\end{array}\right]=\left[\begin{array}{ll}0 &amp; 0 \\ 0 &amp; 0 \\ 0 &amp; 0\end{array}\right]\end{aligned}\)<br/>\(\begin{array}{ll}-\mathrm{a}=0 \Rightarrow \mathrm{a}=0,1-\mathrm{b}=0 \Rightarrow \mathrm{b}=1, \\ 3-\mathrm{c}=0 \Rightarrow \mathrm{c}=3, &amp; 7-\mathrm{d}=0 \Rightarrow \mathrm{d}=7, \\ 5-\mathrm{e}=0 \Rightarrow \mathrm{e}=5, &amp; 6-\mathrm{f}=0 \Rightarrow \mathrm{f}=6,\end{array}\)<br/>\(\therefore D=\left[\begin{array}{ll}0 &amp; 1 \\ 3 &amp; 7 \\ 5 &amp; 6\end{array}\right]\)</div>


tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-aleftbeginarraylll2-0-0-2-2-0-2-2-2endarrayright-then-operatornameadjoperatornameadj-a-is-equal-to

slug:
if-aleftbeginarraylll2-0-0-2-2-0-2-2-2endarrayright-then-operatornameadjoperatornameadj-a-is-equal-to-78181

question:
<div class="question">If \(A=\left[\begin{array}{lll}2 &amp; 0 &amp; 0 \\ 2 &amp; 2 &amp; 0 \\ 2 &amp; 2 &amp; 2\end{array}\right]\), then \(\operatorname{adj}(\operatorname{adj} A)\) is equal to -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(8\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 1 &amp; 1 &amp; 0 \\ 1 &amp; 1 &amp; 1\end{array}\right]\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(16\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 1 &amp; 1 &amp; 0 \\ 1 &amp; 1 &amp; 1\end{array}\right]\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(64\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 1 &amp; 1 &amp; 0 \\ 1 &amp; 1 &amp; 1\end{array}\right]\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(16\left[\begin{array}{lll}1 &amp; 0 &amp; 0 \\ 1 &amp; 1 &amp; 0 \\ 1 &amp; 1 &amp; 1\end{array}\right]\)</span> </div>

answer:
<div class="solution">\(|A|=\left[\begin{array}{lll}<br/>2 &amp; 0 &amp; 0 \\<br/>2 &amp; 2 &amp; 0 \\<br/>2 &amp; 2 &amp; 2<br/>\end{array}\right]=(2)(2)(2)=8\)<br/>Now adj \((\operatorname{adj} A)=|\mathrm{A}|^{3-2} \mathrm{~A}\)<br/>\(=8\left[\begin{array}{lll}<br/>2 &amp; 0 &amp; 0 \\<br/>2 &amp; 2 &amp; 0 \\<br/>2 &amp; 2 &amp; 2<br/>\end{array}\right]=16\left[\begin{array}{ccc}<br/>1 &amp; 0 &amp; 0 \\<br/>1 &amp; 1 &amp; 0 \\<br/>1 &amp; 1 &amp; 1<br/>\end{array}\right]\)</div>


tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-alpha-beta-are-the-roots-of-the-equation-mathrmax2mathrmbxmathrmc0-then-the-roots-of-the-equation-mathrmax2mathrmbxmathrmx1mathrmcmathrmx120-are

slug:
if-alpha-beta-are-the-roots-of-the-equation-mathrmax2mathrmbxmathrmc0-then-the-roots-of-the-equation-mathrmax2mathrmbxmathrmx1mathrmcmathrmx120-are-20599

question:
<div class="question">If \(\alpha, \beta\) are the roots of the equation \(\mathrm{ax}^2+\mathrm{bx}+\mathrm{c}=0\), then the roots of the equation \(\mathrm{ax}^2+\mathrm{bx}(\mathrm{x}+1)+\mathrm{c}(\mathrm{x}+1)^2=0\) are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\alpha-1, \beta-1\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\alpha+1, \beta+1\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\alpha}{\alpha-1}, \frac{\beta}{\beta-1}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\frac{\alpha}{1-\alpha}, \frac{\beta}{1-\beta}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\alpha}{1-\alpha}, \frac{\beta}{1-\beta}\)</span> </div>

answer:
<div class="solution">The second equation can be rewritten as<br/>\(a\left(\frac{x}{x+1}\right)^2+b\left(\frac{x}{x+1}\right)+c=0\)<br/>and hence its roots correspond to \(\frac{\mathrm{x}}{\mathrm{x}+1}=\alpha\) and \(\frac{x}{x+1}=\beta\).<br/>Hence \(x=\frac{\alpha}{1-\alpha}\) and \(\frac{\beta}{1-\beta}\).</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
if-an-electron-has-an-initial-velocity-in-a-direction-different-from-that-of-an-electric-field-the-path-of-the-electron-is

slug:
if-an-electron-has-an-initial-velocity-in-a-direction-different-from-that-of-an-electric-field-the-path-of-the-electron-is-88072

question:
<div class="question">If an electron has an initial velocity in a direction different from that of an electric field, the path of the electron is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a straight line</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a circle</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">an ellipse</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">a parabola</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a parabola</span> </div>

answer:
<div class="solution">The path is a parabola, because initial velocity can be resolved into two rectangular components, one along $\overrightarrow{\mathrm{E}}$ and other $\perp$ to $\overrightarrow{\mathrm{E}}$. The former decreases at a constant rate and latter is unaffected. The resultant path is therefore a parabola.</div>


title: if-and-are-the-roots-of-x-2-x-1-0-then-the-equation-whose-roots-are-100-and-100-are
slug: if-and-are-the-roots-of-x-2-x-1-0-then-the-equation-whose-roots-are-100-and-100-are-81496
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha$ and $\beta$ are the roots of $x^{2}-x+1=0,$ then the equation whose roots are $\alpha^{100}$ and $\beta^{100}$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}-x+1=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2}+x-1=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}-x-1=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x^{2}+x+1=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}+x+1=0$</span> </div>

answer:
<div class="solution">We have $x^{3}+1 \equiv(x+1)\left(x^{2}-x+1\right)$. Therefore, $\alpha$ and $\beta$ are the complex cube roots of -1 so that we may take $\alpha=-\omega$ and $\beta=-\omega^{2},$ where $\omega \neq 1$ is a cube root of unity.
<br/> <br/>Thus $\alpha^{100}=(-\omega)^{100}=\omega$ and
<br/> <br/>$\beta^{100}=\left(-\omega^{2}\right)^{100}=\omega^{2},$ so that the required
<br/> <br/>equation is $\mathrm{x}^{2}+\mathrm{x}+1=0$</div>


title: if-are-the-roots-of-the-equations-x-2-2-x-1-0-then-what-is-the-value-of-2-2-2-2
slug: if-are-the-roots-of-the-equations-x-2-2-x-1-0-then-what-is-the-value-of-2-2-2-2-26271
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">34</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha, \beta$ are the roots of the equations $x^{2}-2 x-1=$ $0,$ then what is the value of $\alpha^{2} \beta^{-2}+\alpha^{-2} \beta^{2}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">30</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">34</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Since, \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2}-2 x-1=0\), then
<br/> <br/>Sum of roots, \(\alpha+\beta=2\) and
<br/> <br/>product of the roots \(\alpha \beta=-1\)
<br/> <br/>Since, \((\alpha+\beta)=\alpha^{2}+\beta^{2}+2 \alpha \beta\)
<br/> <br/>\(\Rightarrow 4=\alpha^{2}+\beta^{2}-2\)
<br/> <br/>\(\Rightarrow \alpha^{2}+\beta^{2}=6\)
<br/> <br/>Now, \(\alpha^{2} \beta^{-2}+\alpha^{-2} \beta^{2}=\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}=\frac{\alpha^{4}+\beta^{4}}{(\alpha \beta)^{2}}\)
<br/> <br/>\(\Rightarrow\left(\alpha^{2}+\beta^{2}\right)^{2}=6^{2}\)
<br/> <br/>\(\Rightarrow \alpha^{4}+\beta^{4}+2 \alpha^{2} \beta^{2}=36\)
<br/> <br/>\(\Rightarrow \alpha^{4}+\beta^{4}+2=36\)
<br/> <br/>\(\Rightarrow \alpha^{4}+\beta^{4}=34 \ldots . .(\mathrm{i})\)
<br/> <br/>\(\Rightarrow \frac{\alpha^{4}+\beta^{4}}{(\alpha \beta)^{2}}=\frac{34}{(-1)^{2}}=34\)
<br/> <br/>[Putting value of \(\alpha^{4}+\beta^{4}=34\) from Equation (i) \(]\)</div>


title: if-arg-leftbarz1rightarg-leftz2right-then
slug: if-arg-leftbarz1rightarg-leftz2right-then-19381
question: <div class="question">If \(\arg \left(\bar{z}_1\right)=\arg \left(z_2\right)\), then</div>
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\mathrm{z}_2=\mathrm{kz}_1^{-1}(\mathrm{k} &gt; 0)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{z}_2=k \mathrm{z}_1(\mathrm{k} &gt; 0)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\left|z_2\right|=\left|\bar{z}_1\right|\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{z}_2=\mathrm{kz}_1^{-1}(\mathrm{k} &gt; 0)\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \bar{z}_1=\frac{z_1 \bar{z}_1}{z_1}=\left|\mathrm{z}_1\right|^2 \mathrm{z}_1^{-1} \\<br/>&amp; \Rightarrow \arg \left(\mathrm{z}_1^{-1}\right)=\arg \left(\bar{z}_1\right) \Rightarrow \arg \left(\mathrm{z}_2\right) \\<br/>&amp; \Rightarrow \mathrm{z}_2=\mathrm{kz}_1^{-1}(\mathrm{k} &gt; 0)<br/>\end{aligned}\)</div>


title: if-be-a-root-of-the-equation-4-x-2-2-x-1-0-then-the-other-root-of-the-equation-is
slug: if-be-a-root-of-the-equation-4-x-2-2-x-1-0-then-the-other-root-of-the-equation-is-95083
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\alpha$ be a root of the equation $4 x^2+2 x-1=0$, then the other root of the equation is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$4 \alpha^3+2 \alpha$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$4 \alpha^2-2 \alpha$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$4 \alpha^3-3 \alpha$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$4 \alpha^3+3 \alpha$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$4 \alpha^3-3 \alpha$
<br/></span> </div>

answer:
<div class="solution">$4 x^2+2 x-1=0$
<br/>$\therefore \quad \alpha+\beta=-\frac{1}{2}, \alpha \beta=-\frac{1}{4}$
<br/>Also, $4 \alpha^2+2 \alpha-1=0$ as $\alpha$ is a root and we have to prove that $\beta=4 \alpha^3-3 \alpha$.
<br/>Now, $4 \alpha^3-3 \alpha=4 \alpha^2 \cdot \alpha-3 \alpha$
<br/>$=\alpha(1-2 \alpha)-3 \alpha=-2 \alpha^2-2 \alpha$
<br/>$=-\frac{1}{2}\left[4 \alpha^2+4 \alpha\right]=-\frac{1}{2}[1-2 \alpha+4 \alpha]$
<br/>$=-\frac{1}{2}(1+2 \alpha)=-\frac{1}{2}-\alpha=\beta$
<br/>Now, $\alpha+\beta=-\frac{1}{2}$
<br/>[from Eq. (i)]
<br/>Hence, $\beta=4 \alpha^3-3 \alpha$.</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
if-binomial-coefficients-of-three-consecutive-terms-of-1-x-2-are-in-h-p-then-the-maximum-value-of-n-is

slug:
if-binomial-coefficients-of-three-consecutive-terms-of-1-x-2-are-in-h-p-then-the-maximum-value-of-n-is-53999

question:
<div class="question">If binomial coefficients of three consecutive terms of $(1+x)^{2}$ are in $H P$, then the maximum value of $\mathrm{n}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let the coefficients of rth, $(r+1)$ th, and (r $+$
<br/> <br/>2) th terms be in HP.
<br/> <br/>Then, $\frac{2}{{ }^{n} C_{r}}=\frac{1}{{ }^{n} C_{r-1}}+\frac{1}{{ }^{n} C_{r}+1}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \quad 2=\frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}}=\frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}+1}} \\
<br/> <br/>\Rightarrow \quad 2=\frac{\mathrm{n}-\mathrm{r}+1}{\mathrm{r}}+\frac{\mathrm{r}+1}{\mathrm{n}-\mathrm{r}} \\
<br/> <br/>\Rightarrow \mathrm{n}^{2}-4 \mathrm{nr}+4 \mathrm{r}^{2}+\mathrm{n}=0 \\
<br/> <br/>\Rightarrow(\mathrm{n}-2 \mathrm{r})^{2}+\mathrm{n}=0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>which is not possible for any value for $n$.</div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
if-c-0-c-1-c-2-c-n-denote-the-binomial-coefficientsin-the-expansion-of-1-x-n-then-the-value-of-c-0-c-0-c-1-c-0-c-1-c-2-c-0-c-1-c-n-1

slug:
if-c-0-c-1-c-2-c-n-denote-the-binomial-coefficientsin-the-expansion-of-1-x-n-then-the-value-of-c-0-c-0-c-1-c-0-c-1-c-2-c-0-c-1-c-n-1-55153

question:
<div class="question">If $\mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots \ldots \ldots \ldots . \mathrm{C}_{\mathrm{n}}$ denote the binomial
<br/> <br/>coefficientsin the expansion of $(1+\mathrm{x})^{\mathrm{n}}$, then the value of $\mathrm{C}_{0}+\left(\mathrm{C}_{0}+\mathrm{C}_{1}\right)+\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}\right)+\ldots$
<br/> <br/>$+\left(C_{0}+C_{1}+\ldots . .+C_{n-1}\right)$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{n} \cdot 2^{\mathrm{n}-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{n} \cdot 2^{\mathrm{n}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(n-1) \cdot 2^{n-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(n-1) \cdot 2^{n}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{n} \cdot 2^{\mathrm{n}-1}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{C}_{0}+\left(\mathrm{C}_{0}+\mathrm{C}_{1}\right)+\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\mathrm{C}_{2}\right)+\ldots \ldots \ldots \ldots \ldots \\ &amp;+\left(\mathrm{C}_{0}+\mathrm{C}_{1}+\ldots \ldots \ldots . \mathrm{C}_{\mathrm{n}-1}\right) \\ &amp;=\mathrm{n} \mathrm{C}_{0}+(\mathrm{n}-1) \mathrm{C}_{1}+(\mathrm{n}-2) \mathrm{C}_{2}+\ldots \ldots \mathrm{C}_{\mathrm{n}-1} \\ &amp;=\mathrm{C}_{1}+2 \mathrm{C}_{2}+3 \mathrm{C}_{3}+4 \mathrm{C}_{4} \ldots \ldots \mathrm{n} \mathrm{C}_{\mathrm{n}}=\mathrm{n} .2^{\mathrm{n}-1} \end{aligned}$</div>


title: if-c-o-s-b-c-o-s-a-n-s-i-n-b-s-i-n-a-m-then-the-value-of-m-2-n-2-sin-2-b-is
slug: if-c-o-s-b-c-o-s-a-n-s-i-n-b-s-i-n-a-m-then-the-value-of-m-2-n-2-sin-2-b-is-30087
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{\cos A}{\cos B}=n, \frac{\sin A}{\sin B}=m$, then the value of $\left(m^{2}-n^{2}\right) \sin ^{2} B$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1+\mathrm{n}^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1-\mathrm{n}^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{n}^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\mathrm{n}^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1-\mathrm{n}^{2}$</span> </div>

answer:
<div class="solution">$\cos A=n \cos B$ and $\sin A=m \sin B$
<br/> <br/>Squaring and adding, we get $1=n^{2} \cos ^{2} B+m^{2} \sin ^{2} B$
<br/> <br/>$$
<br/> <br/>\begin{array}{ll}
<br/> <br/>\Rightarrow &amp; 1=n^{2}\left(1-\sin ^{2} B\right)+m^{2} \sin ^{2} B \\
<br/> <br/>\therefore &amp; \left(m^{2}-n^{2}\right) \sin ^{2} B=1-n^{2}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-complex-number-z-1-z-2-and-0-are-vertices-of-equilateral-triangle-then-z-1-2-z-2-2-z-1-z-2-is-equal-to

slug:
if-complex-number-z-1-z-2-and-0-are-vertices-of-equilateral-triangle-then-z-1-2-z-2-2-z-1-z-2-is-equal-to-21978

question:
<div class="question">If complex number $z_{1}, z_{2}$ and 0 are vertices of equilateral triangle, then $z_{1}^{2}+z_{2}^{2}-z_{1} z_{2}$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$z_{1}-z_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$z_{1}+z_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$z_{1}, z_{2}, 0$ are vertices of an equilateral triangle, so we have $z_{1}^{2}+z_{2}^{2}+0^{2}=z_{1} z_{2}+z_{2} \cdot 0+0 \cdot z_{1}$
<br/> <br/>$$
<br/> <br/>\Rightarrow z_{1}^{2}+z_{2}^{2}=z_{1} z_{2} \Rightarrow z_{1}^{2}+z_{2}^{2}-z_{1} z_{2}=0
<br/> <br/>$$</div>


title: if-cos-1-x-cos-1-2-y-then-4-x-2-4-x-y-cos-y-2-is-equal-to-1
slug: if-cos-1-x-cos-1-2-y-then-4-x-2-4-x-y-cos-y-2-is-equal-to-1-18951
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha$,
<br/> <br/>then $4 x^{2}-4 x y \cos \alpha+y^{2}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \sin 2 \alpha$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4 \sin ^{2} \alpha$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-4 \sin ^{2} \alpha$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \sin ^{2} \alpha$</span> </div>

answer:
<div class="solution">$\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha$
<br/> <br/>$\Rightarrow \cos ^{-1}\left(\frac{x y}{2}+\sqrt{\left(1-x^{2}\right)\left(1-\frac{y^{2}}{4}\right)}\right)=\alpha$
<br/> <br/>$\Rightarrow \cos ^{-1}\left(\frac{x y+\sqrt{4-y^{2}-4 x^{2}+x^{2} y^{2}}}{2}\right)=\alpha$
<br/> <br/>$\Rightarrow 4 x^{2}+y^{2}-4 x y \cos \alpha=4 \sin ^{2} \alpha$</div>


title: if-cos-7-cos-sin-4-then-the-general-value-of-is
slug: if-cos-7-cos-sin-4-then-the-general-value-of-is-95364
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
question:
<div class="question">$\quad$ If $\cos 7 \theta=\cos \theta-\sin 4 \theta,$ then the general value
<br/> <br/>of $\theta$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{n \pi}{6}, \frac{n \pi}{3}+(-1)^{n} \frac{\pi}{18}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{n \pi}{3}, \frac{n \pi}{3}+(-1)^{n} \frac{\pi}{18}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{n \pi}{4}, \frac{n \pi}{3} \pm \frac{\pi}{18}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{n \pi}{4}, \frac{n \pi}{3}+(-1)^{n} \frac{\pi}{18}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{n \pi}{4}, \frac{n \pi}{3}+(-1)^{n} \frac{\pi}{18}$</span> </div>

answer:
<div class="solution">$\cos 7 \theta=\cos \theta-\sin 4 \theta$
<br/> <br/>$\Rightarrow \sin 4 \theta=\cos \theta-\cos 7 \theta$
<br/> <br/>$\Rightarrow \sin 4 \theta=2 \sin 4 \theta \sin 3 \theta$
<br/> <br/>$\Rightarrow \sin 4 \theta(1-2 \sin 3 \theta)=0$
<br/> <br/>$\therefore \sin 4 \theta=0$ or $\sin 3 \theta=\frac{1}{2}$
<br/> <br/>$\Rightarrow 4 \theta=n \pi$ or $3 \theta=n \pi+(-1)^{n} \frac{\pi}{6}$
<br/> <br/>$\Rightarrow \quad \theta=\frac{n \pi}{4}$ or $\frac{n \pi}{3}+(-1)^{n} \frac{\pi}{18}$</div>


title: if-cos-x-y-cos-x-and-cos-x-y-are-in-hp-then-cos-x-sec-y-2-is-equal-to
slug: if-cos-x-y-cos-x-and-cos-x-y-are-in-hp-then-cos-x-sec-y-2-is-equal-to-37215
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo>(</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HP</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mi>x</mi><mi>sec</mi><mo>(</mo><mi>y</mi><mo>/</mo><mn>2</mn><mo>)</mo></math> is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><msqrt><mn>2</mn></msqrt></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>1</mn><mo>/</mo><msqrt><mn>2</mn></msqrt></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><msqrt><mn>2</mn></msqrt></math></span> </div>

answer:
<div class="solution"><p>Given that, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo>(</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mi>x</mi><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HP</mi></math>.</p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>cos</mi><mo>(</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow><mrow><mi>cos</mi><mo>(</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>)</mo><mo>+</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>cos</mi><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>y</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>cos</mi><mi>x</mi><mi>cos</mi><mi>y</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mi>cos</mi><mi>y</mi><mo>=</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>y</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>(</mo><mi>cos</mi><mi>y</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>=</mo><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>y</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mi>y</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>y</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mi>y</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mi>y</mi><mo>)</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mi>y</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>=</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mi>y</mi><mo>⇒</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>=</mo><mn>2</mn><msup><mi>cos</mi><mn>2</mn></msup><mi>y</mi><mo>/</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><msup><mi>sec</mi><mn>2</mn></msup><mo>(</mo><mi>y</mi><mo>/</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>cos</mi><mi>x</mi><mi>sec</mi><mo>(</mo><mi>y</mi><mo>/</mo><mn>2</mn><mo>)</mo><mo>=</mo><mo>±</mo><msqrt><mn>2</mn></msqrt></math></p></div>


tags: ['Mathematics', 'Determinants', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$[z]$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-denotes-the-greatest-integer-less-than-or-equal-to-the-real-number-under-consideration-and-1-x-0-0-y-1-1-z-2-then-the-value-of-the-determinant-is-1

slug:
if-denotes-the-greatest-integer-less-than-or-equal-to-the-real-number-under-consideration-and-1-x-0-0-y-1-1-z-2-then-the-value-of-the-determinant-is-1-48190

question:
<div class="question">If [ ] denotes the greatest integer less than or equal to the real number under consideration and $-1 \leq x&lt;0 ; 0 \leq y&lt;1 ; 1 \leq z&lt;2,$ then the value of the determinant
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/DgEyjvV-MrArf9uTrBueZOFNbrMce-IM0nBGrN4pi28.original.fullsize.png"/><br/> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$[z]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$[y]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$[\mathrm{x}]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Since, $-1 \leq x&lt;0$ $\begin{array}{ll}\therefore \quad &amp; {[\mathrm{x}]=-1} \\ &amp; 0 \leq \mathrm{y}&lt;1 \quad \therefore[\mathrm{y}]=0 \\ &amp; 1 \leq \mathrm{z}&lt;2 \quad \therefore[\mathrm{z}]=1 \\ \therefore \quad &amp; \text { Given determinant }=\left|\begin{array}{ccc}0 &amp; 0 &amp; 1 \\ -1 &amp; 1 &amp; 1 \\ -1 &amp; 0 &amp; 2\end{array}\right|=1=[\mathrm{z}]\end{array}$</div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$[z]$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-denotes-the-greatest-integer-less-than-or-equal-to-the-real-number-under-consideration-and-1-x-0-0-y-1-1-z-2-then-the-value-of-the-determinant-is

slug:
if-denotes-the-greatest-integer-less-than-or-equal-to-the-real-number-under-consideration-and-1-x-0-0-y-1-1-z-2-then-the-value-of-the-determinant-is-41888

question:
<div class="question">If [ ] denotes the greatest integer less than or equal to the real number under consideration and $-1 \leq x&lt;0 ; 0 \leq y&lt;1 ; 1 \leq z&lt;2,$ then the value of the determinant
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/DgEyjvV-MrArf9uTrBueZOFNbrMce-IM0nBGrN4pi28.original.fullsize.png"/><br/> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$[z]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$[y]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$[\mathrm{x}]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Since, $-1 \leq x&lt;0$ $\begin{array}{ll}\therefore \quad &amp; {[\mathrm{x}]=-1} \\ &amp; 0 \leq \mathrm{y}&lt;1 \quad \therefore[\mathrm{y}]=0 \\ &amp; 1 \leq \mathrm{z}&lt;2 \quad \therefore[\mathrm{z}]=1 \\ \therefore \quad &amp; \text { Given determinant }=\left|\begin{array}{ccc}0 &amp; 0 &amp; 1 \\ -1 &amp; 1 &amp; 1 \\ -1 &amp; 0 &amp; 2\end{array}\right|=1=[\mathrm{z}]\end{array}$</div>


title: if-e-1-and-e-2-are-the-eccentricities-of-a-hyperbola-3-x-2-3-y-2-25-and-its-conjugate-then
slug: if-e-1-and-e-2-are-the-eccentricities-of-a-hyperbola-3-x-2-3-y-2-25-and-its-conjugate-then-45513
tags: ['Mathematics', 'Hyperbola', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>2</mn></msub></math> are the eccentricities of a hyperbola <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>25</mn></math> and its conjugate, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>e</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>e</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>e</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>e</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mn>4</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub><mo>+</mo><msub><mi>e</mi><mn>2</mn></msub><mo>=</mo><mn>4</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub><mo>+</mo><msub><mi>e</mi><mn>2</mn></msub><mo>=</mo><msqrt><mn>2</mn></msqrt></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>e</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>e</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mn>4</mn></math></span> </div>

answer:
<div class="solution"><p>Given equation can be written as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>25</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msub><mi>e</mi><mn>1</mn></msub><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mn>1</mn></msqrt><mo>=</mo><msqrt><mn>2</mn></msqrt></math></p><p>The equation of conjugate hyperbola is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>25</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msub><mi>e</mi><mn>2</mn></msub><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac></msqrt><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mn>1</mn></msqrt><mo>=</mo><msqrt><mn>2</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msubsup><mi>e</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>e</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><msup><mfenced><msqrt><mn>2</mn></msqrt></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msqrt><mn>2</mn></msqrt></mfenced><mn>2</mn></msup><mo>=</mo><mn>4</mn></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>e</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>e</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><mn>4</mn></math>.</p></div>


title: if-e-3-x-e-x-e-5-x-a-0-a-1-x-a-2-x-2-a-3-x-3-then-the-value-of-2-a-1-2-3-a-3-2-5-a-5-is
slug: if-e-3-x-e-x-e-5-x-a-0-a-1-x-a-2-x-2-a-3-x-3-then-the-value-of-2-a-1-2-3-a-3-2-5-a-5-is-45452
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\frac{e^{x}+e^{5 x}}{e^{3 x}}=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\ldots$ then
<br/> <br/>the value of $2 a_{1}+2^{3} a_{3}+2^{5} a_{5}+\ldots$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$e^{2}+e^{-2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$e^{4}-e^{-4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$e^{t}+e^{-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let $\frac{\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{5 \mathrm{x}}}{\mathrm{e}^{3 \mathrm{x}}}=\mathrm{a}_{0}+\mathrm{a}_{1} \mathrm{x}+\mathrm{a}_{2} \mathrm{x}^{2}+\mathrm{a}_{3} \mathrm{a}^{3}+\ldots .$
<br/> <br/>$\begin{aligned} &amp;=\frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{e}^{3 \mathrm{x}}}+\frac{\mathrm{e}^{5 \mathrm{x}}}{\mathrm{e}^{3 \mathrm{x}}}=\mathrm{a}_{0}+\mathrm{a}_{1} \mathrm{x}+\mathrm{a}_{2} \mathrm{x}^{2}+\ldots . \\=&amp; \mathrm{e}^{-2 \mathrm{x}}+\mathrm{e}^{2 \mathrm{x}}=\mathrm{a}_{0}+\mathrm{a}_{1} \mathrm{x}+\mathrm{a}_{2} \mathrm{x}^{2}+\mathrm{a}_{3} \mathrm{x}^{3}+\ldots . \end{aligned}$
<br/> <br/>By using $\mathrm{e}^{\mathrm{x}}=1+\mathrm{x}+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{3}}{3 !}+{ }_{--}--$ and
<br/> <br/>$\mathrm{e}^{-\mathrm{x}}=1-\mathrm{x}+\frac{\mathrm{x}^{2}}{2 !}-\frac{\mathrm{x}^{3}}{3 !}+---$
<br/> <br/>$\mathrm{e}^{-2 \mathrm{x}}+\mathrm{e}^{2 \mathrm{x}}=2\left[1+\frac{(2 \mathrm{x})^{2}}{21}+\frac{(2 \mathrm{x})^{4}}{41}+\ldots .\right]$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} a^{3}+\ldots . \\
<br/> <br/>=a_{1}=a_{3}=a_{5}=\ldots=0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Hence, $2 \mathrm{a}_{1}+2^{3} \mathrm{a}_{3}+2^{5} \mathrm{a}_{5}+\ldots .=0$</div>


title: if-e-and-f-are-events-such-that-0-p-f-1-then
slug: if-e-and-f-are-events-such-that-0-p-f-1-then-73642
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{E}$ and $\mathrm{F}$ are events such that $0 &lt; \mathrm{P}(\mathrm{F}) &lt; 1$, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$P(E \mid F)+P(\bar{E} \mid F)=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$P(E \mid F)+P(E \mid \bar{F})=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$P(\bar{E} \mid F)+P(E \mid \bar{F})=1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$P(E \mid \bar{F})+P(\bar{E} \mid \bar{F})=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$P(E \mid F)+P(\bar{E} \mid F)=1$</span> </div>

answer:
<div class="solution">$P(E \mid F)+P(\bar{E} \mid F)$<br/>$=\frac{P(E \cap F)+P(\bar{E} \cap F)}{P(F)}=\frac{P((E \cup \bar{E}) \cap F)}{P(F)}=\frac{P(F)}{P(F)}=1$</div>


tags: ['Physics', 'Electromagnetic Waves', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-e-and-k-represent-electric-field-and-propagation-vectors-of-the-em-waves-in-vacuum-then-magnetic-field-vector-is-given-by-angular-frequency

slug:
if-e-and-k-represent-electric-field-and-propagation-vectors-of-the-em-waves-in-vacuum-then-magnetic-field-vector-is-given-by-angular-frequency-22871

question:
<div class="question">If $\overrightarrow{\mathrm{E}}$ and $\overrightarrow{\mathrm{K}}$ represent electric field and propagation vectors of the EM waves in vacuum, then magnetic field vector is given by: ( $\omega$ - angular frequency) :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\omega}(\overrightarrow{\mathrm{K}} \times \overrightarrow{\mathrm{E}})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\omega(\overrightarrow{\mathrm{E}} \times \overrightarrow{\mathrm{K}})$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\omega(\overrightarrow{\mathrm{K}} \times \overrightarrow{\mathrm{E}})$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\overrightarrow{\mathrm{K}} \times \overrightarrow{\mathrm{E}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\omega}(\overrightarrow{\mathrm{K}} \times \overrightarrow{\mathrm{E}})$</span> </div>

answer:
<div class="solution">For an EM wave<br/>$\overrightarrow{\mathrm{B}}=\frac{1}{\omega}(\overrightarrow{\mathrm{K}} \times \overrightarrow{\mathrm{E}})$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
title:
if-enthalpies-of-formation-of-c-2-h-4-g-co-2-g-and-h-2-o-l-at-2-5-c-and-1-atm-pressure-are-52394-and-286-kj-mol-respectively-the-change-in-enthalpy-fo

slug:
if-enthalpies-of-formation-of-c-2-h-4-g-co-2-g-and-h-2-o-l-at-2-5-c-and-1-atm-pressure-are-52394-and-286-kj-mol-respectively-the-change-in-enthalpy-fo-37613

question:
<div class="question">If enthalpies of formation of $\mathrm{C}_2 \mathrm{H}_4(\mathrm{~g}), \mathrm{CO}_2(\mathrm{~g})$ and $\mathrm{H}_2 \mathrm{O}(\mathrm{l})$ at $25^{\circ} \mathrm{C}$ and $1 \mathrm{~atm}$ pressure are 52,394 and $-286 \mathrm{~kJ} / \mathrm{mol}$ respectively, the change in enthalpy for combustion of $\mathrm{C}_2 \mathrm{H}_4$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-141.2 \mathrm{~kJ} / \mathrm{mol}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> $-1412 \mathrm{~kJ} / \mathrm{mol}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$+14.2 \mathrm{~kJ} / \mathrm{mol}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$+1412 \mathrm{~kJ} / \mathrm{mol}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> $-1412 \mathrm{~kJ} / \mathrm{mol}$</span> </div>

answer:
<div class="solution">Enthalpy of formation of $\mathrm{C}_2 \mathrm{H}_4, \mathrm{CO}_2$ and $\mathrm{H}_2 \mathrm{O}$ are $52,-394$ and $-286 \mathrm{~kJ} / \mathrm{mol}$ respectively.<br/>(Given)<br/>The reaction is<br/>$\mathrm{C}_2 \mathrm{H}_4+3 \mathrm{O}_2 \rightarrow 2 \mathrm{CO}_2+2 \mathrm{H}_2 \mathrm{O}$<br/>change in enthalpy,<br/>$(\Delta H)=\Delta H_{\text {products }}-\Delta H_{\text {reactants }}$<br/>$=2 \times(-394)+2 \times(-286)-(52+0)$<br/>$=-1412 \mathrm{~kJ} / \mathrm{mol}$.</div>


title: if-equation-10-x-5-2-10-y-4-2-2-3-x-4-y-1-2-represents-a-hyperbola-then
slug: if-equation-10-x-5-2-10-y-4-2-2-3-x-4-y-1-2-represents-a-hyperbola-then-82745
tags: ['Mathematics', 'Hyperbola', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mi>x</mi><mo>-</mo><mn>5</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>10</mn><mi>y</mi><mo>-</mo><mn>4</mn><msup><mo>)</mo><mn>2</mn></msup><mo>=</mo><msup><mi>λ</mi><mn>2</mn></msup><mo>(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>-</mo><mn>1</mn><msup><mo>)</mo><mn>2</mn></msup></math> represents a hyperbola, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>&lt;</mo><mi>λ</mi><mo>&lt;</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>&gt;</mo><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>&lt;</mo><mo>-</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>&gt;</mo><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>λ</mi><mo>&lt;</mo><mn>2</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>&lt;</mo><mo>-</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>&gt;</mo><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Given, equation of hyperbola is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>10</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>10</mn><mi>y</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>λ</mi><mn>2</mn></msup><msup><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> can be rewritten as</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow></mfenced><mn>2</mn></msup></msqrt><mfenced close="|" open="|"><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>-</mo><mn>1</mn></mrow><mn>5</mn></mfrac></mfenced></mfrac><mo>=</mo><mfenced close="|" open="|"><mfrac><mi>λ</mi><mn>2</mn></mfrac></mfenced></math></p><p>This is of the form of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>P</mi><mi>S</mi></mrow><mrow><mi>P</mi><mi>M</mi></mrow></mfrac><mo>=</mo><mi>e</mi></math></p><p>Where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is any point on the hyperbola and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> is a focus and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> is the point of directrix.</p><p>Here, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mfrac><mi>λ</mi><mn>2</mn></mfrac></mfenced><mo>&gt;</mo><mn>1</mn><mo>⇒</mo><mfenced close="|" open="|"><mi>λ</mi></mfenced><mo>&gt;</mo><mn>2</mn><mo>  </mo><mfenced><mrow><mo>∵</mo><mi>e</mi><mo>&gt;</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>λ</mi><mo>&lt;</mo><mo>-</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>&gt;</mo><mn>2</mn></math></p></div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
if-equation-of-transverse-wave-is-y-x-0-cos-2-nt-x-maximum-velocity-of-particle-is-twice-of-wave-velocity-if-is

slug:
if-equation-of-transverse-wave-is-y-x-0-cos-2-nt-x-maximum-velocity-of-particle-is-twice-of-wave-velocity-if-is-97403

question:
<div class="question">If equation of transverse wave is $y=x_{0} \cos$ $2 \pi\left(\mathrm{nt}-\frac{\mathrm{x}}{\lambda}\right) .$ Maximum velocity of particle is twice of wave velocity, if $\lambda$ is-</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi / 2 \mathrm{x}_{0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \pi x_{0}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pi$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\pi x_{0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\pi x_{0}$</span> </div>

answer:
<div class="solution">$y=x_{0} \cos 2 \pi\left(\right.$ nt $\left.-\frac{x}{\lambda}\right)$
<br/> <br/>$\mathrm{y}=\mathrm{x}_{0} \cos \frac{2 \pi}{\lambda}(\mathrm{vt}-\mathrm{x}) \quad[\because \mathrm{v}=\mathrm{n} \lambda]$
<br/> <br/>$\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)_{\max }=\mathrm{x}_{0} \times \frac{2 \pi}{\lambda} \mathrm{v}=2 \mathrm{v}($ given $) \therefore \lambda=\pi \mathrm{x}_{0}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-f-and-g-are-differentiable-function-in-0-1-satisfying-f-0-2-g-1-g-0-0-and-f-1-6-then-for-some-c-0-1

slug:
if-f-and-g-are-differentiable-function-in-0-1-satisfying-f-0-2-g-1-g-0-0-and-f-1-6-then-for-some-c-0-1-13625

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are differentiable function in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2</mn><mo>=</mo><mi>g</mi><mfenced><mn>1</mn></mfenced><mo>,</mo><mo> </mo><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>6</mn></math>, then for some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>∈</mo><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><mi>g</mi><mfenced><mi>c</mi></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><mn>3</mn><msup><mi>g</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><msup><mi>g</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><mn>2</mn><msup><mi>g</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><mn>2</mn><msup><mi>g</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced></math></span> </div>

answer:
<div class="solution"><p>Let</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>2</mn><mi>g</mi><mfenced><mi>x</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>-</mo><mn>2</mn><mi>g</mi><mfenced><mn>0</mn></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2</mn><mo>-</mo><mn>2</mn><mfenced><mn>0</mn></mfenced></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2</mn><mo>,</mo><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>-</mo><mn>2</mn><mi>g</mi><mfenced><mn>1</mn></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>6</mn><mo>-</mo><mn>2</mn><mo>(</mo><mn>2</mn><mo>)</mo></math>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>6</mn><mo>,</mo><mi>g</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>2</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>h</mi><mfenced><mn>1</mn></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>2</mn><msup><mi>g</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></math></p><p>By Rolle's theorem <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>-</mo><mn>2</mn><msup><mi>g</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced><mo>=</mo><mn>2</mn><msup><mi>g</mi><mo>'</mo></msup><mfenced><mi>c</mi></mfenced></math></p></div>


title: if-f-is-an-even-function-and-g-is-an-odd-function-then-the-function-fog-is
slug: if-f-is-an-even-function-and-g-is-an-odd-function-then-the-function-fog-is-44643
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}$ is an even function and $\mathrm{g}$ is an odd function, then the function fog is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">an even function</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">an odd function</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">neither even nor odd</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">a periodic function</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">an even function</span> </div>

answer:
<div class="solution">We have, $f o g(-x)=f[g(-x)]=f[-g(x)]$ $(\because g$ is odd $)$
<br/>$\begin{array}{l}
<br/>=\mathrm{f}[\mathrm{g}(\mathrm{x})] \quad(\because \mathrm{fis} \text { even }) \\
<br/>=\operatorname{fog}(\mathrm{x}) \forall \mathrm{x} \in \mathrm{R}
<br/>\end{array}$
<br/>$\therefore$ fog is an even function.</div>


title: if-f-x-1-cos-2-x-2-then-the-value-of-f-2
slug: if-f-x-1-cos-2-x-2-then-the-value-of-f-2-41422
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\sqrt{1+\cos ^2\left(\mathrm{x}^2\right)}$, then the value of $f^{\prime}\left(\frac{\sqrt{\pi}}{2}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{\pi}}{6}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\sqrt{\frac{\pi}{6}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\sqrt{6}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{\sqrt{6}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\sqrt{\frac{\pi}{6}}$</span> </div>

answer:
<div class="solution"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/_NTjqRT8hKJ5THfUNdP4T3sWIILr9OmymI6oIXX4-EQ.original.fullsize.png"/><br/><br/>On differentiating (i) w.r.t.x, we get<br/>$f^{\prime}(x)=\frac{-2 \sin x^2 \cos x^2}{\sqrt{1+\cos ^2 x^2}}(x)$<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/EdpMz33XoOBGACcfTnTZM5cxVVi4UlbFmlVTJTu-wOM.original.fullsize.png"/><br/><br/>Put, $x=\frac{\sqrt{\pi}}{2}$ in (ii), we get<br/>$\begin{aligned}<br/>&amp; \mathrm{f}^{\prime}\left(\frac{\sqrt{\pi}}{2}\right)=-\frac{\sqrt{\pi}}{2} \cdot \frac{\sin 2\left(\frac{\pi}{4}\right)}{\sqrt{1+\frac{1}{2}}} \\<br/>&amp; =-\frac{\sqrt{\pi}}{2} \cdot \frac{\sin \frac{\pi}{2}}{\sqrt{\frac{3}{2}}}=-\sqrt{\frac{\pi}{6}}<br/>\end{aligned}$</div>


title: if-f-x-1-x-2-x-then-fof-of-x-is
slug: if-f-x-1-x-2-x-then-fof-of-x-is-27489
question: <div class="question">If $f(x)=\frac{x}{\sqrt{1+x^2}}$, then (fof of) $(x)$ is</div>
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3 x}{\sqrt{1+x^2}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{x}{\sqrt{1+3 x^2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3 x}{\sqrt{1-x^2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x}{\sqrt{1+3 x^2}}$</span> </div>

answer:
<div class="solution">$f(x)=\frac{x}{\sqrt{1+x^2}}$<br/>fof$=\frac{\frac{x}{\sqrt{1+x^2}}}{\sqrt{1+\frac{x^2}{1+x^2}}}=\frac{x}{\sqrt{2 x^2+1}}$<br/>fofof $=\frac{\frac{x}{\sqrt{2 x^2+1}}}{\sqrt{1+\frac{x^2}{2 x^2+1}}}=\frac{x}{\sqrt{1+3 x^2}}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">[-1,1]</span> </div>
source_db: MarksBatch2_P1.db
title:
if-f-x-a-2-1-a-2-1-x-3-3-x-5-is-a-decreasing-function-of-x-in-r-then-the-set-of-possible-values-of-a-independent-of-x-is

slug:
if-f-x-a-2-1-a-2-1-x-3-3-x-5-is-a-decreasing-function-of-x-in-r-then-the-set-of-possible-values-of-a-independent-of-x-is-59198

question:
<div class="question">If $\mathrm{f}(\mathrm{x})=\frac{\mathrm{a}^{2}-1}{\mathrm{a}^{2}+1} \mathrm{x}^{3}-3 \mathrm{x}+5$ is a decreasing
<br/>function of $\mathrm{x}$ in $\mathbf{R}$, then the set of possible values of a (independent of $\mathrm{x}$ ) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(1, \infty)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-\infty,-1)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">[-1,1]</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$f^{\prime}(x)=3\left(\frac{a^{2}-1}{a^{2}+1}\right) x^{2}-3$
<br/>$\mathrm{f}^{\prime}(\mathrm{x})&lt;0$ for all $\mathrm{x}$ if $\mathrm{a}^{2}-1 \leq 0 \Rightarrow-1 \leq \mathrm{a} \leq 1$</div>


title: if-f-x-cos-x-g-x-cos-2-x-h-x-cos-3-x-and-i-x-tan-x-then-which-of-the-following-option-is-correct
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
slug:
if-f-x-cos-x-g-x-cos-2-x-h-x-cos-3-x-and-i-x-tan-x-then-which-of-the-following-option-is-correct-12973

question:
<div class="question">If $f(x)=\cos x, g(x)=\cos 2 x, h(x)=\cos 3 x$ and $I(x)$ $=\tan x$, then which of the following option is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f(x)$ and $g(x)$ are strictly decreasing in $(0, \pi /$ 2)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$h(x)$ is neither increasing nor decreasing in $(0, \pi / 2)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{I}(\mathrm{x})$ is strictly increasing in $(0, \pi / 2)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>All are correct</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>All are correct</span> </div>

answer:
<div class="solution">(a) Let $f(x)=\cos x$, then $f^{\prime}(x)=-\sin x$.<br/>In interval $\left(0, \frac{\pi}{2}\right), f^{\prime}(x) &lt; 0$<br/>Therefore, $f(x)$ is strictly decreasing on $\left(0, \frac{\pi}{2}\right)$<br/>(b) Let $f(x)=\cos 2 x \Rightarrow f^{\prime}(x)=-2 \sin x 2 x$<br/>In interval $\left(0, \frac{\pi}{2}\right), f^{\prime}(x) &lt; 0$<br/>Because $\sin 2 x$ will either lie in the first or second quadrant which will give a positive value.<br/>Therefore, $f(x)$ is strictly decreasing on $\left(0, \frac{\pi}{2}\right)$<br/>(c) Let $f^{\prime}(x)=\cos 3 x$<br/>$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-3 \sin 3 \mathrm{x}$. In Interval $\left(0, \frac{\pi}{3}\right), \mathrm{f}^{\prime}(\mathrm{x}) &lt; 0$<br/>Because $\sin 3 \mathrm{x}$ will either lie in the first or second quadrant which will give a positive value.<br/>Therefore, $\mathrm{f}(\mathrm{x})$ irstrictly decreasing on $\left(0, \frac{\pi}{3}\right)$.<br/>When $\mathrm{x} \in\left(\frac{\pi}{3}, \frac{\pi}{2}\right)$, then $\mathrm{f}^{\prime}(\mathrm{x})&gt;0$<br/>Because $\sin 3 x$ will lie in the third quadrant. Therefore, $f(x)$ is not strictly decreasing on<br/>$\left(0, \frac{\pi}{2}\right)$<br/>(d) Let $f(x)=\tan x \Rightarrow f^{\prime}(x)=\sec ^2 x$.<br/>In Interval $\mathrm{x} \in\left(0, \frac{\pi}{2}\right), \mathrm{f}^{\prime}(\mathrm{x})&gt;0$<br/>Therefore, $f(x)$ is not strictly decreasing on<br/>$\left(0, \frac{\pi}{2}\right)$</div>


title: if-f-x-e-x-x-0-1-x-x-0-then-2
slug: if-f-x-e-x-x-0-1-x-x-0-then-2-38711
tags: ['Mathematics', 'Continuity and Differentiability', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced close="" open="{"><mtable columnalign="center left"><mtr><mtd><msup><mi>e</mi><mi>x</mi></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>≤</mo><mn>0</mn></mtd></mtr><mtr><mtd><mfenced close="|" open="|"><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>x</mi><mo>&gt;</mo><mn>0</mn></mtd></mtr></mtable></mfenced></math>, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced close="" open="{"><mtable><mtr><mtd><msup><mi>e</mi><mi>x</mi></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>≤</mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mi>x</mi><mo>,</mo></mtd><mtd><mn>0</mn><mo>&lt;</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>-</mo><mn>1</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo>&gt;</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e0a76017-cdf5-467b-8de4-e4bb94a4a2a7-image.png" style="width: 392px; height: 200px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mi>H</mi><mi>D</mi><mo>:</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mfenced><mrow><mn>0</mn><mo>+</mo><mi>h</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced><mn>0</mn></mfenced></mrow><mi>h</mi></mfrac><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mo>-</mo><mfenced close="]" open="["><mfrac><mrow><mi>h</mi><mo>-</mo><mn>1</mn></mrow><mi>h</mi></mfrac></mfenced><mo>=</mo><mo>-</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mi>H</mi><mi>D</mi><mo>:</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mfenced><mrow><mn>0</mn><mo>-</mo><mi>h</mi></mrow></mfenced><mo>-</mo><mi>f</mi><mfenced><mn>0</mn></mfenced></mrow><mrow><mo>-</mo><mi>h</mi></mrow></mfrac><mo>=</mo><munder><mi>lim</mi><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mi>e</mi><mrow><mo>-</mo><mi>h</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow><mrow><mo>-</mo><mi>h</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math></p><p>So, it is not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.</p><p>Similarly, it is not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> but it is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>.</p></div>


title: if-f-x-f-x-0-1-f-x-d-x-f-0-1-then-f-x
slug: if-f-x-f-x-0-1-f-x-d-x-f-0-1-then-f-x-65849
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>d</mi><mi>x</mi><mo>,</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo></math></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>e</mi></mrow><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>e</mi><mi>x</mi></msup><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>e</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>e</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mn>2</mn><mo>-</mo><mi>e</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>e</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>e</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mn>2</mn><mo>-</mo><mi>e</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>e</mi></mrow><mrow><mn>3</mn><mo>+</mo><mi>e</mi></mrow></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>e</mi></mrow><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p>Given,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>d</mi><mi>x</mi><mo>,</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>f</mi><mrow><mo>'</mo><mo>'</mo></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mrow><msup><mi>f</mi><mrow><mo>'</mo><mo>'</mo></mrow></msup><mfenced><mi>x</mi></mfenced></mrow><mrow><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></mrow></mfrac><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>∫</mo><mfrac><mrow><msup><mi>f</mi><mrow><mo>'</mo><mo>'</mo></mrow></msup><mfenced><mi>x</mi></mfenced></mrow><mrow><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mo>∫</mo><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>log</mi><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>A</mi><msup><mi>e</mi><mi>x</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>A</mi><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>K</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>A</mi><mo>+</mo><mi>K</mi><mo>=</mo><mn>1</mn></math>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>ii</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>A</mi><msup><mi>e</mi><mi>x</mi></msup><mo>=</mo><mi>A</mi><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>K</mi><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfenced><mrow><mi>A</mi><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>k</mi><mo>+</mo><msubsup><mfenced close="]" open="["><mrow><mi>A</mi><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>K</mi><mi>x</mi></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>k</mi><mo>+</mo><mi>A</mi><mi>e</mi><mo>-</mo><mi>A</mi><mo>+</mo><mi>K</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>A</mi><mo>(</mo><mi>e</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>2</mn><mi>k</mi><mo>=</mo><mn>0</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>iii</mi><mo>)</mo></math></p><p>From Eqs. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ii</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>iii</mi><mo>)</mo></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac><mo>,</mo><mi>k</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>e</mi></mrow><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>e</mi><mi>x</mi></msup></mrow><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>e</mi></mrow><mrow><mn>3</mn><mo>-</mo><mi>e</mi></mrow></mfrac></math></p></div>


title: if-f-x-l-o-g-1-x-2-x-l-o-g-c-o-s-x-0-x-0-x-0-then-f-x-is
slug: if-f-x-l-o-g-1-x-2-x-l-o-g-c-o-s-x-0-x-0-x-0-then-f-x-is-10955
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=\left\{\begin{array}{cl}\frac{x \log \cos x}{\log \left(1+x^{2}\right)} &amp; , x \neq 0 \\ 0 &amp; , x=0\end{array}\right.$ then $f(x)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">continuous as well as differentiable at $\mathrm{x}=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">continuous but not differentiable at $x=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">differentiable but not continuous at $\mathrm{x}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">neither continuous nor differentiable at $\mathrm{x}=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">continuous as well as differentiable at $\mathrm{x}=0$</span> </div>

answer:
<div class="solution">We have,
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\mathrm{Lf}^{\prime}(0) &amp;=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0-\mathrm{h})-\mathrm{f}(0)}{-\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{-\mathrm{h} \log \cosh }{-\mathrm{h} \log \left(1+\mathrm{h}^{2}\right)} \\
<br/> <br/>=&amp; \lim _{\mathrm{h} \rightarrow 0} \frac{\log \cosh }{\log \left(1+\mathrm{h}^{2}\right)}\left(\frac{0}{0} \text { form }\right)
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>$\begin{aligned} &amp;=\lim _{\mathrm{h} \rightarrow 0} \frac{-\tan \mathrm{h}}{2 \mathrm{~h} /\left(1+\mathrm{h}^{2}\right)}=-1 / 2 \\ \mathrm{Rf}^{\prime}(0) &amp;=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0+\mathrm{h})-\mathrm{f}(0)}{\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{h} \log \cosh }{\mathrm{h} \log \left(1+\mathrm{h}^{2}\right)} \\ &amp;=\lim _{\mathrm{h} \rightarrow 0} \frac{\log \cos \mathrm{h}}{\log \left(1+\mathrm{h}^{2}\right)}\left(\frac{0}{0} \text { form }\right) \\ &amp;=\lim _{\mathrm{h} \rightarrow 0} \frac{-\tan \mathrm{h}}{2 \mathrm{~h} /\left(1+\mathrm{h}^{2}\right)}=\frac{-1}{2} \end{aligned}$
<br/> <br/>Since $\mathrm{Lf}^{\prime}(0)=\operatorname{Rf}^{\prime}(0),$ therefore $\mathrm{f}(\mathrm{x})$ is dif-
<br/> <br/>ferentiable at $x=0$
<br/> <br/>Since differentiability $\Rightarrow$ continuity, there-fore $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$.</div>


tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-f-x-x-2-1-0-x-2-2-x-3-2-x-3-then-the-quadratic-equation-whose-roots-are-lim-x-2-f-x-and-lim-x-2-f-x-is

slug:
if-f-x-x-2-1-0-x-2-2-x-3-2-x-3-then-the-quadratic-equation-whose-roots-are-lim-x-2-f-x-and-lim-x-2-f-x-is-99460

question:
<div class="question">If $f(x)=\left\{\begin{array}{l}x^2-1,0 &lt; x &lt; 2 \\ 2 x+3,2 \leq x &lt; 3\end{array}\right.$, then the quadratic equation whose roots are $\lim _{x \rightarrow 2^{-}} f(x)$ and $\lim _{x \rightarrow 2^{+}} f(x)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^2-6 x+9=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^2-7 x+8=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^2-14 x+49=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x^2-10 x+21=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^2-10 x+21=0$</span> </div>

answer:
<div class="solution">Since, $f(x)= \begin{cases}x^2-1, &amp; 0 &lt; x &lt; 2 \\ 2 x+3, &amp; 2 \leq x &lt; 3\end{cases}$<br/>Now, $\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{-}}\left(x^2-1\right)$<br/>$\begin{aligned}<br/>&amp; =\lim _{h \rightarrow 0}\left[(2-h)^2-1\right]=\lim _{h \rightarrow 0}\left[4+h^2-4 h-1\right] \\<br/>&amp; =\lim _{h \rightarrow 0}\left[h^2-4 h+3\right]=0-0+3=3<br/>\end{aligned}$<br/>and $\lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{+}}(2 x+3)$<br/>$=\lim _{h \rightarrow 0}[2(2+h)+3]=\lim _{h \rightarrow 0}[4+2 h+3]=7$<br/>Hence the quadratic equation whose roots are 3 and 7 is given by $x^2-(3+7) x+(3+7)=0 \Rightarrow x^2$ $-10 x+21=0$</div>


title: if-f-x-x-2-2-x-15-x-2-3-x-10-a-when-x-5-when-x-5-is-continuous-at-x-5-then-the-value-of-a-will-be
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
slug:
if-f-x-x-2-2-x-15-x-2-3-x-10-a-when-x-5-when-x-5-is-continuous-at-x-5-then-the-value-of-a-will-be-89429

question:
<div class="question">If $f(x)=\left\{\begin{array}{cl}\frac{x^{2}+3 x-10}{x^{2}+2 x-15} &amp; \text { , when } x \neq-5 \\ \text { a }, &amp; \text { when } x=-5\end{array}\right.$
<br/> <br/>is continuous at $x=-5$, then the value of 'a' will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 / 2$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$7 / 8$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 / 7$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 / 3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7 / 8$</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow-5} f(x)=\frac{(x-2)(x+5)}{(x+5)(x-3)}=\frac{-7}{-8}=\frac{7}{8}$</div>


title: if-f-x-x-lo-g-x-and-f-0-0-then-the-value-of-for-which-rolles-theorem-can-be-applied-in-01-is
slug: if-f-x-x-lo-g-x-and-f-0-0-then-the-value-of-for-which-rolles-theorem-can-be-applied-in-01-is-21981
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $f(x)=x^{\alpha} \log x$ and $f(0)=0,$ then the value of $\alpha$ for which Rolle's theorem can be applied in [0,1] is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1 / 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 2$</span> </div>

answer:
<div class="solution">For Rolle's theorem in $[\mathrm{a}, \mathrm{b}], \mathrm{f}(\mathrm{a})=\mathrm{f}(\mathrm{b}),$
<br/>$\operatorname{In}[0,1] \Rightarrow \mathrm{f}(0)=\mathrm{f}(1)=0$
<br/>$\because$ the function has to be continuous in [0,1] $\Rightarrow f(0)=\lim _{x \rightarrow 0^{+}} f(x)=0$
<br/>$\Rightarrow \lim _{x \rightarrow 0} x^{\alpha} \log x=0 \Rightarrow \lim _{x \rightarrow 0} \frac{\log x}{x^{-\alpha}}=0$
<br/>Applying L.H. Rule $\lim _{x \rightarrow 0} \frac{1 / x}{-\alpha x^{-\alpha-1}}=0$
<br/>$\Rightarrow \lim _{x \rightarrow 0} \frac{-x^{\alpha}}{\alpha}=0 \Rightarrow \alpha&gt;0$</div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
title:
if-force-f-length-l-and-time-t-are-assumed-to-be-fundamental-units-then-the-dimensional-formula-of-the-mass-will-be

slug:
if-force-f-length-l-and-time-t-are-assumed-to-be-fundamental-units-then-the-dimensional-formula-of-the-mass-will-be-70276

question:
<div class="question">If force $(F)$, length $(L)$ and time $(T)$ are assumed to be fundamental units, then the dimensional formula of the mass will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{FL}^{-1} \mathrm{~T}^{2}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{FL}^{-1} \mathrm{~T}^{-2}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{FL}^{-1} \mathrm{~T}^{-1}\right]$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{FL}^{2} \mathrm{~T}^{2}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{FL}^{2} \mathrm{~T}^{2}\right]$</span> </div>

answer:
<div class="solution">Let $\mathrm{m}=\mathrm{KF}^{\mathrm{a}} \mathrm{L}^{\mathrm{b}} \mathrm{T}^{\mathrm{c}}$
<br/> <br/>Substituting the dimensions of
<br/> <br/>$[\mathrm{F}]=\left[\mathrm{MLT}^{-2}\right],[\mathrm{L}]=[\mathrm{L}]$ and $[\mathrm{T}]=[\mathrm{T}]$ and comparing both side, we get $\mathrm{m}=\mathrm{FL}^{-1} \mathrm{~T}^{2}$</div>


title: if-frac1i-alpha1i-alphaai-b-then-a2b2-equals-to
slug: if-frac1i-alpha1i-alphaai-b-then-a2b2-equals-to-58467
question: <div class="question">If \(\frac{1-i \alpha}{1+i \alpha}=A+i B\), then \(A^2+B^2\) equals to</div>
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\alpha^2\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-\alpha^2\)</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; A+i B=\frac{1-i \alpha}{1+i \alpha} \Rightarrow A-i B=\frac{1+i \alpha}{1-i \alpha} \\<br/>&amp; \Rightarrow(A+i B)(A-i B)=\frac{(1-i \alpha)(1+i \alpha)}{(1+i \alpha)(1-i \alpha)}=1 \\<br/>&amp; \Rightarrow A^2+B^2=1<br/>\end{aligned}\)</div>


title: if-frac2-x35-frac4-x12-then-x-lies-in-the-interval
slug: if-frac2-x35-frac4-x12-then-x-lies-in-the-interval-17531
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(\frac{2 x+3}{5}  &lt; \frac{4 x-1}{2}\), then \(x\) lies in the interval</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\left[0, \frac{11}{16}\right)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\left[\frac{11}{16}, \infty\right)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\left(0, \frac{11}{16}\right)\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\left(\frac{11}{16}, \infty\right)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left(\frac{11}{16}, \infty\right)\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \frac{2 x+3}{5}  &lt; \frac{4 x-1}{2} \Rightarrow-16 x &lt; -11 \\<br/>&amp; \Rightarrow 16 x &gt; 11 \Rightarrow x &gt; \frac{11}{16} \\<br/>&amp; \text { Hence, } x \in\left(\frac{11}{16}, \infty\right)<br/>\end{aligned}\)</div>


title: if-ftfrac1t1t-then-fprime1-t-is-equal-to
slug: if-ftfrac1t1t-then-fprime1-t-is-equal-to-78270
question: <div class="question">If \(f(t)=\frac{1-t}{1+t}\), then \(f^{\prime}(1 / t)\) is equal to</div>
tags: ['Mathematics', 'Mathematical Reasoning', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{(1+t)^2}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{(t-1)^2}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{-2 t^2}{(t+1)^2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{2}{(t-1)^2}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{-2 t^2}{(t+1)^2}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; f^{\prime}(t)=\frac{d}{d t}\left[\frac{1-t}{1+t}\right]=\frac{(1+t)(-1)-(1-t) \times(1)}{(1+t)^2} \\<br/>&amp; =\frac{-1-t-1+t}{(1+t)^2}=\frac{-2}{(1+t)^2}<br/>\end{aligned}\)<br/>\(f^{\prime}[1 / t]=\frac{-2}{\left(1+\frac{1}{t}\right)^2}=\frac{-2 t^2}{(t+1)^2}\)</div>


title: if-fx-is-a-function-that-is-odd-and-even-simultaneously-then-mathrmf3mathrmf2-is-equal-to
slug: if-fx-is-a-function-that-is-odd-and-even-simultaneously-then-mathrmf3mathrmf2-is-equal-to-81857
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(f(x)\) is a function that is odd and even simultaneously, then \(\mathrm{f}(3)-\mathrm{f}(2)\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">\(\mathrm{f}(\mathrm{x})=0 ~\forall~ \mathrm{x} \in \mathrm{R} \Rightarrow \mathrm{f}(3)-\mathrm{f}(2)=0\)</div>


title: if-fxfrac1x1x-the-domain-of-f1x-is
slug: if-fxfrac1x1x-the-domain-of-f1x-is-70063
question: <div class="question">If \(f(x)=\frac{1-x}{1+x}\) the domain of \(f^{-1}(x)\) is</div>
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{R}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{R}-\{-1\}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\((-\infty,-1)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\((-1, \infty)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{R}-\{-1\}\)</span> </div>

answer:
<div class="solution">Let \(f(x)=y\). Then \(\frac{1-x}{1+x}=y\)<br/>\(\Rightarrow x=\frac{1-y}{1+y} \Rightarrow f^{-1}(y)=\frac{1-y}{1+y}\)<br/>Thus, \(f^{-1}(x)=\frac{1-x}{1+x}\)<br/>Clearly, \(\mathrm{f}^{-1}(\mathrm{x})\) is defined for \(1+\mathrm{x} \neq 0\).<br/>Hence, domain of \(\mathrm{f}^{-1}(\mathrm{x})\) is \(\mathrm{R}-\{-1\}\)</div>


title: if-fxfracxsqrt1x2-then-fof-of-x-is
slug: if-fxfracxsqrt1x2-then-fof-of-x-is-97224
question: <div class="question">If \(f(x)=\frac{x}{\sqrt{1+x^2}}\), then (fof of \()(x)\) is</div>
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{3 x}{\sqrt{1+x^2}}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{\mathrm{x}}{\sqrt{1+3 \mathrm{x}^2}}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{3 x}{\sqrt{1-x^2}}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\mathrm{x}}{\sqrt{1+3 \mathrm{x}^2}}\)</span> </div>

answer:
<div class="solution">\(\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\sqrt{1+\mathrm{x}^2}}\)<br/>\((f o f)(x)=\frac{\frac{x}{\sqrt{1+x^2}}}{\sqrt{1+\frac{x^2}{1+x^2}}}=\frac{x}{\sqrt{2 x^2+1}}\)<br/>\((fofof)(x)=\frac{\frac{x}{\sqrt{2 x^2+1}}}{\sqrt{1+\frac{x^2}{2 x^2+1}}}=\frac{x}{\sqrt{1+3 x^2}}\)</div>


tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
if-fxleftbeginarraylx-text-when-x-text-is-rational-1x-text-when-x-text-is-irrational-endarrayright-then-fof-x-is-given-as

slug:
if-fxleftbeginarraylx-text-when-x-text-is-rational-1x-text-when-x-text-is-irrational-endarrayright-then-fof-x-is-given-as-69548

question:
<div class="question">If \(f(x)=\left\{\begin{array}{l}x ; \text { when } x \text { is rational } \\ 1-x ; \text { when } x \text { is irrational }\end{array}\right.\), then fof \((x)\) is given as</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{x}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1+\mathrm{x}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{x}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; f o f(x)=\left\{\begin{array}{l}<br/>f(x) ; \text { when } f(x) \text { is rational } \\<br/>1-f(x) \text {; when } f(x) \text { is irrational }<br/>\end{array}\right. \\<br/>&amp; =\left\{\begin{array}{l}<br/>\mathrm{x} ; \quad \text { when } \mathrm{x} \text { is rational } \\<br/>1-(1-\mathrm{x}) ; \text { when } \mathrm{x} \text { is irrational }<br/>\end{array}=\mathrm{x}\right. \\<br/>&amp;<br/>\end{aligned}\)</div>


title: if-g-is-the-inverse-of-function-f-and-f-x-sin-x-then-g-x-is-equal-to
slug: if-g-is-the-inverse-of-function-f-and-f-x-sin-x-then-g-x-is-equal-to-76728
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $g$ is the inverse of function $f$ and $f^{\prime}(x)=\sin x$, then $\mathrm{g}^{\prime}(\mathrm{x})$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\operatorname{cosec}\{\mathrm{g}(\mathrm{x})\}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sin \{g(x)\}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\sin \{g(x)\}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\sin \{g(x)\}}$</span> </div>

answer:
<div class="solution">Since, $g$ is the inverse of function $f$. Therefore, $\mathrm{g}(\mathrm{x})=\mathrm{f}^{-1}(\mathrm{x})$
<br/> <br/>$\Rightarrow \mathrm{f}[\mathrm{g}(\mathrm{x})]=\mathrm{x}$
<br/> <br/>$\Rightarrow$ fog $(\mathrm{x})=\mathrm{x},$ for all $\mathrm{x}$
<br/> <br/>Differentiate both side,w.r.tx
<br/> <br/>$\Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\{\operatorname{fog}(\mathrm{x})\}=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x}),$ for all $\mathrm{x}$
<br/> <br/>$\Rightarrow \mathrm{f}^{\prime}[\mathrm{g}(\mathrm{x})] \mathrm{g}^{\prime}(\mathrm{x})=1,$ for all $\mathrm{x}$
<br/> <br/>$\Rightarrow \sin \{\mathrm{g}(\mathrm{x})\} \mathrm{g}^{\prime}(\mathrm{x})=1,$ for all $\mathrm{x}$
<br/> <br/>(By defn of $\left.f^{\prime}(x)\right)$
<br/> <br/>$\Rightarrow \mathrm{g}^{\prime}(\mathrm{x})=\frac{1}{\sin \{\mathrm{g}(\mathrm{x})\}}$</div>


title: if-g-x-x-2-x-2-and-2-1-g-o-f-x-2-x-2-5-x-2-then-f-x-is-equal-to
slug: if-g-x-x-2-x-2-and-2-1-g-o-f-x-2-x-2-5-x-2-then-f-x-is-equal-to-94014
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $g(x)=x^2+x-2$ and $\frac{1}{2} g o f(x)=2 x^2-5 x+2$ then $f(x)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$2 x-3$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$2 x+3$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$2 x^2+3 x+1$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$2 x^2-3 x+1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$2 x-3$
<br/></span> </div>

answer:
<div class="solution">We have, $\frac{1}{2} go f(x)=2 x^2-5 x+2$
<br/>$\begin{aligned}
<br/>&amp; \Rightarrow g(f(x))=4 x^2-10 x+4 \\
<br/>&amp; \Rightarrow(f(x))^2+f(x)-2=4 x^2-10 x+4 \\
<br/>&amp; \Rightarrow(f(x))^2+f(x)-\left(4 x^2-10 x+6\right)=0 \\
<br/>&amp; \Rightarrow f(x)=\frac{-1 \pm \sqrt{1+4\left(4 x^2-10 x+6\right)}}{2} \\
<br/>&amp; \Rightarrow f(x)=\frac{-1 \pm \sqrt{16 x^2-40 x+25}}{2} \\
<br/>&amp; \Rightarrow f(x)=\frac{-1 \pm(4 x-5)}{2}=2 x-3,-2 x+2
<br/>\end{aligned}$
<br/>Hence, $f(x)=2 x-3$</div>


tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-h-is-the-altitude-of-a-parallelopiped-determined-by-the-vectors-a-b-c-and-the-base-is-taken-to-be-the-parallelogram-determined-by-a-and-b-where-a-i

slug:
if-h-is-the-altitude-of-a-parallelopiped-determined-by-the-vectors-a-b-c-and-the-base-is-taken-to-be-the-parallelogram-determined-by-a-and-b-where-a-i-88109

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is the altitude of a parallelopiped determined by the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>^</mo></mover><mo>,</mo><mo> </mo><mover><mi>b</mi><mo>^</mo></mover><mo>,</mo><mo> </mo><mover><mi>c</mi><mo>^</mo></mover></math> and the base is taken to be the parallelogram determined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>^</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>b</mi><mo>^</mo></mover></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>^</mo></mover><mo>=</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover><mo>,</mo><mo> </mo><mover><mi>b</mi><mo>^</mo></mover><mo>=</mo><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>4</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>-</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>c</mi><mo>^</mo></mover><mo>=</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover><mo>,</mo></math> then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn><msup><mi>h</mi><mn>2</mn></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> </div>

answer:
<div class="solution"><p>Volume of the parallelopiped <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="|" open="|"><mfenced close="]" open="["><mrow><mover><mi>a</mi><mo>^</mo></mover><mo> </mo><mover><mi>b</mi><mo>^</mo></mover><mo> </mo><mover><mi>c</mi><mo>^</mo></mover></mrow></mfenced></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math> (area of the base parallelogram) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><mi>h</mi><mo>=</mo><mfenced close="|" open="|"><mfenced close="]" open="["><mrow><mover><mi>a</mi><mo>^</mo></mover><mo> </mo><mover><mi>b</mi><mo>^</mo></mover><mo> </mo><mover><mi>c</mi><mo>^</mo></mover></mrow></mfenced></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced close="|" open="|"><mrow><mover><mi>a</mi><mo>^</mo></mover><mo>×</mo><mover><mi>b</mi><mo>^</mo></mover></mrow></mfenced><mi>h</mi><mo>=</mo><mfenced close="|" open="|"><mfenced close="]" open="["><mrow><mover><mi>a</mi><mo>^</mo></mover><mo> </mo><mover><mi>b</mi><mo>^</mo></mover><mo> </mo><mover><mi>c</mi><mo>^</mo></mover></mrow></mfenced></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced close="|" open="|"><mrow><mo>-</mo><mn>5</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mi>h</mi><mo>=</mo><mfenced close="|" open="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mtable><mtr></mtr><mtr></mtr><mtr></mtr></mtable></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msqrt><mn>38</mn></msqrt><mi>h</mi><mo>=</mo><mn>4</mn><mo>⇒</mo><mi>h</mi><mo>=</mo><mfrac><mn>4</mn><msqrt><mn>38</mn></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>38</mn><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mn>4</mn><mn>2</mn></msup><mo>⇒</mo><mn>19</mn><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><mn>8</mn></math></p><p>Hence, option (c) is correct.</p></div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 v$</span> </div>
source_db: MarksBatch2_P1.db
title:
if-hydrogen-atom-an-electron-jumps-from-bigger-orbit-to-smaller-orbit-so-that-radius-of-smaller-orbit-is-onefourth-of-radius-of-bigger-orbit-if-speed-

slug:
if-hydrogen-atom-an-electron-jumps-from-bigger-orbit-to-smaller-orbit-so-that-radius-of-smaller-orbit-is-onefourth-of-radius-of-bigger-orbit-if-speed-50831

question:
<div class="question">If hydrogen atom, an electron jumps from bigger orbit to smaller orbit so that radius of smaller orbit is one-fourth of radius of bigger orbit. If speed of electron in bigger orbit was $\mathrm{v}$, then speed in smaller orbit is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{v}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{v}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$v$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2 v$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Radius of the orbit, $\mathrm{r}_{\mathrm{n}} \propto \mathrm{n}^{2}$
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>&amp; \frac{\mathrm{r}_{\mathrm{nbig}}}{\mathrm{r}_{\mathrm{n} \text { small }}}=\frac{\mathrm{n}_{\text {big }}^{2}}{\mathrm{n}_{\text {small }}^{2}}=\frac{4}{1} \\
<br/> <br/>\Rightarrow &amp; \frac{\mathrm{n}_{\text {big }}}{\mathrm{n}_{\text {small }}}=2 \\
<br/> <br/>\Rightarrow &amp; \frac{\mathrm{n}_{\text {small }}}{\mathrm{n}_{\text {big }}}=\frac{1}{2}
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>Velocity of electron in $\mathrm{n}^{\text {th }}$ orbit
<br/> <br/>$$
<br/> <br/>\begin{array}{c}
<br/> <br/>\quad \mathrm{v}_{\mathrm{n}} \propto \frac{1}{\mathrm{n}} \\
<br/> <br/>\frac{\mathrm{v}_{\mathrm{n} \mathrm{big}}}{\mathrm{v}_{\mathrm{n} \text { small }}}=\frac{\mathrm{n}_{\text {small }}}{\mathrm{n}_{\text {big }}}=\frac{1}{2} \\
<br/> <br/>\Rightarrow \mathrm{v}_{\mathrm{n} \text { small }}=2\left(\mathrm{v}_{\mathrm{n} \mathrm{big}}\right)=2 \mathrm{v}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: if-in-a-a-bc-2-b-2-a-2-c-2-then-s-i-n-b-s-i-n-3-b-is-equal-to
slug: if-in-a-a-bc-2-b-2-a-2-c-2-then-s-i-n-b-s-i-n-3-b-is-equal-to-89398
tags: ['Mathematics', 'Properties of Triangles', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">If in a $\triangle A B C, 2 b^2=a^2+c^2$, then $\frac{\sin 3B}{\sin B}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{c^2-a^2}{2 c a}$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{c^2-a^2}{c a}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{\left(c^2-a^2\right)^2}{(c a)^2}$
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>$\left[\frac{c^2-a^2}{2 c a}\right]^2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\left[\frac{c^2-a^2}{2 c a}\right]^2$</span> </div>

answer:
<div class="solution">$\frac{\sin 3 B}{\sin B}=\frac{3 \sin B-4 \sin ^3 B}{\sin B}$
<br/>$=3-4 \sin ^2 B$
<br/>$=3-4\left(1-\cos ^2 B\right)$
<br/>$=-1+\frac{4\left(a^2+c^2-b^2\right)^2}{4(a c)^2}$
<br/>$=-1+\frac{\left(\frac{a^2+c^2}{2}\right)^2}{(a c)^2}$
<br/>$=-1+\frac{\left(a^2+c^2\right)^2}{4(a c)^2}$
<br/>$=(\frac{c^2-a^2}{2 a c})^2$</div>


title: if-in-a-binomial-distribution-n-4-p-x-0-81-16-then-p-x-4-equals
slug: if-in-a-binomial-distribution-n-4-p-x-0-81-16-then-p-x-4-equals-23270
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If in a binomial distribution $\mathrm{n}=4, \mathrm{P}(\mathrm{X}=0)=\frac{16}{81}$, then $\mathrm{P}(\mathrm{X}=4)$ equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{16}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{81}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{27}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{8}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{81}$</span> </div>

answer:
<div class="solution">Given $n=4$ and $P(X=0)=\frac{16}{81}$
<br/> <br/>Let $\mathrm{p}$ be the probability of success and $\mathrm{q}$ that of failure in a trial.
<br/> <br/>$$
<br/> <br/>\text { Then, } \mathrm{P}(\mathrm{X}=0)={ }^{4} \mathrm{C}_{0} \mathrm{p}^{0} \mathrm{q}^{4}=\frac{16}{81}
<br/> <br/>$$
<br/> <br/>$\Rightarrow \quad \mathrm{q}^{4}=\left(\frac{2}{3}\right)^{4}$
<br/> <br/>$\left(\because{ }^{n} C_{0}=1\right)$
<br/> <br/>$\Rightarrow \quad \mathrm{q}=\frac{2}{3} \Rightarrow \mathrm{p}=\frac{1}{3}$
<br/> <br/>$\therefore \quad \mathrm{P}(\mathrm{X}=4)={ }^{4} \mathrm{C}_{4} \mathrm{p}^{4} \mathrm{q}^{0}=\mathrm{p}^{4}=\left(\frac{1}{3}\right)^{4}=\frac{1}{81}$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
if-in-the-expansion-of-left2xfrac14xrightn-t37-t2-and-sum-of-the-binomial-coefficients-of-second-and-third-terms-is-36-then-the-value-of-x-is

slug:
if-in-the-expansion-of-left2xfrac14xrightn-t37-t2-and-sum-of-the-binomial-coefficients-of-second-and-third-terms-is-36-then-the-value-of-x-is-71708

question:
<div class="question">If in the expansion of \(\left(2^x+\frac{1}{4^x}\right)^n, T_3=7 T_2\) and sum of the binomial coefficients of second and third terms is 36 , then the value of \(x\) is -</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(-1 / 3\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(-1 / 2\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(1 / 3\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(1 / 2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-1 / 3\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; { }^n C_1+{ }^n C_2=36 \Rightarrow n=8 \\<br/>&amp; T_3=7 \mathrm{~T}_2 \Rightarrow\left(2^{\mathrm{x}}\right)^3=1 / 2 \\<br/>&amp; 3 \mathrm{x}=-1 \Rightarrow \mathrm{x}=-\frac{1}{3}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
if-int-frac2-cos-xsin-xlambdacos-xsin-x2-d-x-a-ln-cos-xsin-x2b-xc-then-the-ordered-triplet-a-b-lambda-is

slug:
if-int-frac2-cos-xsin-xlambdacos-xsin-x2-d-x-a-ln-cos-xsin-x2b-xc-then-the-ordered-triplet-a-b-lambda-is-85992

question:
<div class="question">If \(\int \frac{2 \cos x-\sin x+\lambda}{\cos x+\sin x-2} d x\) \(=A \ln |\cos x+\sin x-2|+B x+C\).<br/>Then the ordered triplet \(A, B, \lambda\) is -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\left(\frac{1}{2}, \frac{3}{2},-1\right)\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\left(\frac{3}{2}, \frac{1}{2},-1\right)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\left(\frac{1}{2},-1, \frac{3}{2}\right)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\left(\frac{3}{2},-1, \frac{1}{2}\right)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left(\frac{3}{2}, \frac{1}{2},-1\right)\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{A} \ln |\cos \mathrm{x}+\sin \mathrm{x}-2|+\mathrm{Bx}+\mathrm{C}) \\<br/>&amp;= \mathrm{A} \frac{\cos \mathrm{x}-\sin \mathrm{x}}{\cos \mathrm{x}+\sin \mathrm{x}-2}+\mathrm{B} \\<br/>&amp;= \frac{\mathrm{A} \cos \mathrm{x}-\mathrm{A} \sin \mathrm{x}+\mathrm{B} \cos \mathrm{x}+\mathrm{B} \sin \mathrm{x}-2 \mathrm{~B}}{\cos \mathrm{x}+\sin \mathrm{x}-2} \\<br/>&amp; \therefore 2=\mathrm{A}+\mathrm{B} \text { or }-1=-\mathrm{A}+\mathrm{B} ; \lambda=-2 \mathrm{~B} \\<br/>&amp; \therefore \mathrm{A}=3 / 2, \mathrm{~B}=1 / 2, \lambda=-1<br/>\end{aligned}\)</div>


title: if-int0nx-mathrmdx66-then-n
slug: if-int0nx-mathrmdx66-then-n-44038
question: <div class="question">If \(\int_0^n[x] \mathrm{dx}=66\), then \(n=\)</div>
tags: ['Mathematics', 'Pair of Lines', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">9</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \int_0^n[x] d x=\int_0^1 0 d x+\int_1^2 1 d x+\int_2^3 2 d x+\ldots \ldots+ \\<br/>&amp; \int_{n-1}^n(n-1) d x \\<br/>&amp; =1+2+3+\ldots .+(n-1)=\frac{(n-1) n}{2}=66 \\<br/>&amp; \Rightarrow n(n-1)=132 \Rightarrow n=12<br/>\end{aligned}\)</div>


title: if-is-square-root-of-identity-matrix-of-order-2-then
slug: if-is-square-root-of-identity-matrix-of-order-2-then-99232
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left[\begin{array}{cc}\alpha &amp; \beta \\ \gamma &amp; -\alpha\end{array}\right]$ is square root of identity matrix of order 2 then -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1+\alpha^{2}+\beta \gamma=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1+\alpha^{2}-\beta \gamma=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1-\alpha^{2}+\beta \gamma=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\alpha^{2}+\beta \gamma=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha^{2}+\beta \gamma=1$</span> </div>

answer:
<div class="solution">$\left[\begin{array}{cc}\alpha &amp; \beta \\ \gamma &amp; -\alpha\end{array}\right]=\sqrt{\mathrm{I}_{2}}$;
<br/> <br/>$\left[\begin{array}{lc}\alpha &amp; \beta \\ \gamma &amp; -\alpha\end{array}\right]\left[\begin{array}{cc}\alpha &amp; \beta \\ \gamma &amp; -\alpha\end{array}\right]=\left[\begin{array}{cc}1 &amp; 0 \\ 0 &amp; 1\end{array}\right]$
<br/> <br/>$\Rightarrow \alpha^{2}+\beta \gamma=1$</div>


title: if-is-the-angle-between-the-vectors-4-i-k-and-i-j-k-then-sin-cos-equals-to
slug: if-is-the-angle-between-the-vectors-4-i-k-and-i-j-k-then-sin-cos-equals-to-90715
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is the angle between the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover><mo>,</mo></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>sin</mi><mi>θ</mi><mo>+</mo><mi>cos</mi><mi>θ</mi></mrow></mfenced></math> equals to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p>Let </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>→</mo></mover><mo>=</mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>b</mi><mo>→</mo></mover><mo>=</mo><mn>4</mn><mfenced><mrow><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced></math></p><p>Then,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>a</mi><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mn>3</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi>b</mi><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>0</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><mn>4</mn><msqrt><mn>2</mn></msqrt></math></p><p>Now,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><mover><mi>a</mi><mo>→</mo></mover><mo>·</mo><mover><mi>b</mi><mo>→</mo></mover></mrow><mrow><mfenced close="|" open="|"><mover><mi>a</mi><mo>→</mo></mover></mfenced><mfenced close="|" open="|"><mover><mi>b</mi><mo>→</mo></mover></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>cos</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover><mo>+</mo><mover><mi mathvariant="normal">k</mi><mo>^</mo></mover></mrow></mfenced><mo>·</mo><mfenced><mrow><mn>4</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>4</mn><mover><mi>k</mi><mo>^</mo></mover></mrow></mfenced></mrow><mrow><mfenced close="|" open="|"><mover><mi>a</mi><mo>→</mo></mover></mfenced><mfenced close="|" open="|"><mover><mi>b</mi><mo>→</mo></mover></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>cos</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>-</mo><mn>4</mn></mrow><mrow><mfenced close="|" open="|"><mover><mi>a</mi><mo>→</mo></mover></mfenced><mfenced close="|" open="|"><mover><mi>b</mi><mo>→</mo></mover></mfenced></mrow></mfrac><mo>=</mo><mn>0</mn></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>θ</mi><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>θ</mi></msqrt><mo>=</mo><mn>1</mn></math></p><p>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>θ</mi><mo>+</mo><mi>cos</mi><mi>θ</mi><mo>=</mo><mn>1</mn></math></p></div>


title: if-is-the-complex-cube-root-of-unity-then-the-value-of-2-1-8-3-32-9-128-27-is
slug: if-is-the-complex-cube-root-of-unity-then-the-value-of-2-1-8-3-32-9-128-27-is-49758
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\omega$ is the complex cube root of unity, then the value of $\omega+\omega^{\left(\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\frac{27}{128}+\ldots\right)}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\mathrm{i}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">i</span> </li> </ul>

answer:
<div class="solution">Consider $\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\frac{27}{128}+\ldots .$
<br/> <br/>Which can be written as
<br/> <br/>$$
<br/> <br/>\frac{3^{0}}{2^{1}}+\frac{3^{1}}{2^{3}}+\frac{3^{2}}{2^{5}}+\frac{3^{3}}{2^{7}}+\ldots .
<br/> <br/>$$ $=\frac{1}{2}\left[1+\frac{3}{2^{2}}+\frac{3^{2}}{2^{4}}+\frac{3^{3}}{2^{6}}+\ldots .\right]$
<br/> <br/>Since $\left(1+\frac{3}{2^{2}}+\frac{3^{2}}{2^{4}}+_{--}-\right)$ is a G.P.
<br/> <br/>therfore by sum of infinite G.P, we have
<br/> <br/>$$
<br/> <br/>=\frac{1}{2}\left[\frac{1}{1-\frac{3}{2^{2}}}\right]=\frac{1}{2}\left[\frac{1}{1-\frac{3}{4}}\right]=2
<br/> <br/>$$
<br/> <br/>$\therefore \quad$ Given expression
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=-1 \\
<br/> <br/>{\left[\because 1+\omega+\omega^{2}=0\right]}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: if-k-sin-1-x-cos-1-x-tan-1-x-k-then
slug: if-k-sin-1-x-cos-1-x-tan-1-x-k-then-65545
question: <div class="question">If $k \leq \sin ^{-1} x+\cos ^{-1} x+\tan ^{-1} x \leq K,$ then $-$</div>
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{k}=0, \mathrm{~K}=\pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{k}=0, \mathrm{~K}=\pi / 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{k}=\pi / 2, \mathrm{~K}=\pi$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\sin ^{-1} x+\cos ^{-1} x+\tan ^{-1} x=\frac{\pi}{2}+\tan ^{-1} x$
<br/>Since domain of the function $\mathrm{x} \in[-1,1]$ $\therefore-\frac{\pi}{4} \leq \tan ^{-1} x \leq \frac{\pi}{4}$
<br/>Hence, $\mathrm{k}=\frac{\pi}{4}$ and $\mathrm{K}=\frac{3 \pi}{4}$</div>


title: if-lim-n-a-n-1-n-2-1-n-b-where-a-is-finite-number-then
slug: if-lim-n-a-n-1-n-2-1-n-b-where-a-is-finite-number-then-68877
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mi>a</mi><mi>n</mi><mo>-</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><mi>n</mi></mrow></mfrac></mrow></mfenced><mo>=</mo><mi>b</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is finite number, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfrac><mrow><mo>an</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>)</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></mrow></mfenced></mrow><mrow><mn>1</mn><mo>+</mo><mi>n</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfrac><mrow><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mo>∞</mo></math>,</p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>-</mo><mn>1</mn><mo>≠</mo><mn>0</mn></math> limit does not exist and if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, then</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfrac><mrow><mi>a</mi><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>a</mi><mo>=</mo><mi>b</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>1</mn></math></p></div>


title: if-lim-x-x-sin-x-1-a-and-lim-x-0-x-sin-x-1-b-then-which-one-of-the-following-is-correct
slug: if-lim-x-x-sin-x-1-a-and-lim-x-0-x-sin-x-1-b-then-which-one-of-the-following-is-correct-36279
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\lim _{x \rightarrow} x \sin \left(\frac{1}{x}\right)=A$ and $\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)=B$,
<br/> <br/>then which one of the following is correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$A=1$ and $B=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad A=0$ and $B=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$A=0$ and $B=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$A=1$ and $B=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A=1$ and $B=0$</span> </div>

answer:
<div class="solution">As given,
<br/> <br/>$$
<br/> <br/>\mathrm{A}=\lim _{\mathrm{x} \rightarrow \infty} \mathrm{x} \sin \left(\frac{1}{\mathrm{x}}\right)=\lim _{\mathrm{x} \rightarrow \infty} \frac{\sin \left(\frac{1}{\mathrm{x}}\right)}{\left(\frac{1}{\mathrm{x}}\right)}
<br/> <br/>$$
<br/> <br/>Let $\mathrm{t}=\frac{1}{\mathrm{x}}$ when $\mathrm{x} \rightarrow \alpha, \mathrm{t} \rightarrow 0$
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\Rightarrow \mathrm{A} &amp;=\lim _{t \rightarrow \infty} \frac{\sin \mathrm{t}}{\mathrm{t}}=1 \\
<br/> <br/>&amp;\left[\because \lim _{\mathrm{t} \rightarrow 0} \frac{\sin \mathrm{x}}{\mathrm{x}}=1\right]
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>and $\mathrm{B}=\lim _{\mathrm{x} \rightarrow 0} \mathrm{x} \sin \left(\frac{1}{\mathrm{x}}\right)$
<br/> <br/>$$
<br/> <br/>\Rightarrow \mathrm{B}=\lim _{\mathrm{x} \rightarrow 0} \mathrm{x} \cdot \lim _{\mathrm{x} \rightarrow 0} \sin \left(\frac{1}{\mathrm{x}}\right)
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\Rightarrow \mathrm{B}=0
<br/> <br/>$$
<br/> <br/>$\therefore \quad A=1$ and $B=0$ is correct</div>


